InternalBallistics 10262675 - Outros (2025)

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ERRATA.Page 69, line 9 from top, for w read q.after evolved”read per unit of wei ght.1 1 from top, for P,0, q read Po v q.7 frombottom,for“v V0 read “ 0q6 frombottom, for 10f w read 10f q.70, lines5 and 9 from top and 9 frombottom, for“f to read f q .84, line 2 from top, for“P read P,.8 from top, read2 ”10 from top, dale there.14 fromtop, for a”read w.21 from top, for N‘ 3 xi ,”read N a:i.3 fromtop, after“ table insert12 frombottom, for g r 7 .8 frombottom, for B read flw a l l a “ , M wnfiseScattered over a. number of serial publications in Franmany of which are not accessible to the general body 0artillerists. M. Sarrau, with that spirit of liberality Whiocharacterises the true man of science , has most kindly giveme permission to make use of his investigations, and it is tPREFA CE.IN my treatise On the Application ofWire to the Construcof Ordnance ,’ published in 1884, I touched lightly on oneor two questions relating to Internal Ballistics, such aschambering, slow-burning powder, and heat imparted tothe gun.Shortly afterwards I presented a paper to the Institutionof Civil Engineers on Guns considered as ThermodynamicMachines,” which was published in the Minutes 01 Proceedings,’ vol . lxxx.,1884—85 and in 1887, I printed a smallpamphlet on Internal Ballistics,which, however, was onlyc irculated among a few friends.The subject of Internal Ballistics appears to have met withcomparatively li ttle attention in this country, and althoughthe researches of Dr. Hutton are very valuable, they, owingto the change of conditions, are inapplicable to a great extentto the present time .The researches of French artillerists of late years have sheda flood of light on the subject of the action of gunpowder,and espec ially those of M. Emile Sarrau, but these arescattered over a number of serial publications in Franc e ,many of which are not accessible to the general body ofartillerists. M. Sarrau, with that spirit of liberality whichcharacterises the true man of science, has most kindly givenme permission to make use of his investigations, and it is tovi PREFACE.this that the most valuable part of the following treatiseis due .In ChapterI . I have briefly treatediofExplosives in general .Chapter II. treats more particularly of Fired Gunpowder,the nature of the Products of Combustion,of Ignition andCombustion, the influence of the Form of Gram, the Temperature of Combustion, the Strength of Powder, the Loss ofTemperature by the cooling action of the metal of the gun,and the Pressure and Movement of the Products of Combustion.Chapter III. is devoted to M. Sarrau’s investigations of theFormulas for MuzzleVelocity and MaximumPressure .Chapter IV. contains a few remarks on the Designing ofGuns, and on Pressure Curves .Chapter V. treats of Guns as Thermodynamic Machines.I am fully sensible of the many imperfections of thepresent treatise, and of my own incompetency to treat thisimportant subject in an exhaustive manner, but I am notwithout hope that it may be found useful and suggestive tothose who are interested in artillery questions. My objecthas been, to the best of my ability, to combine theoreticalinvestigations with practical utility.In the Report of the Royal Commission on Warlike Storespresided over by Sir James Stephen, a distinctionis made between what is there called the sc ience ofgunnery, and the science of gun construction, and I amrepresented as c laiming the latter as my special sc ience . Inever did anything of the kind. I certainly claimed tohave a special knowledge of the subject of the application ofwire to gun construction, but I did not, and could not,represent gunnery as one science and gun construction as another.What I tried to show to the Commission, but apparentlyfailed in, was that gun construction should be conducted onFEBEACE. viiand guided by, scientific knowledge, and that such knowledge greatly depended ou these theoretical considerations.The Commissioners appeared to doubt whether it is possibleto state prec isely the relation in which theory and practiceought to stand to eachother, and in this they were supported by Sir Frederick Bramwell, who gave itas his opinion,that it would be dangerous to give theorists control oversuch a matter as the manufacture of a gun.It is a grave error to suppose that theory and sound practice are, or can be, divergent. Hypothesis and practice may ,and very often do, disagree, but theory never, unless it be afalse theory. De Quincey says, Theory is,in fact, no morethan a system of laws, abstracted from experience ; consequently, if any apparent contradiction should exist betweenthem, this could only argue that the theory had been falselyor imperfectly abstracted; in which case the sensible inference would be, not a summons to forego theories, but a callfor better or mo re enlarged theories.”AndKant, in his essayOn the common saying, that such and such a thing maybe true in theory, but does not hold good in practice,” says,It is far more tolerable that an unleamed person shouldrepresent theory as superfluous for the purpose of his imaginary practice, than that a shallow refiner, whilst concedingthe value of theory for speculative and scholastic uses, shouldcouple with this concession the doctrine, that in practice thecase is otherwise ; and that upon coming out of the schoolsinto the world, aman will be made sensible of having pursuedmere philosophic dreams. In short, that what sounds wellin theory is not merely superfluous, but absolutely false forpractice. Now the practical engineer who should expresshimself in such terms upon the scienceofmechanics, or theartillery ofiicer who should say of the doctrine of projectiles,that the theory of it was conceived indeed with great subviii PREFACE.tilty, but was of little practical value, because in the actualexperience of the art it was found that the experimentalresults did not conform to the theory, would expose themselves to derision. For, supposing that in the first caseshould be superadded to the Theory of Mechanics that ofFriction, and that in the second, to the Theory of Projectiles were superadded that of the resistance of the airwhich in effect amounts to this, that if, instead of rejectingtheory, still more theory were added— in that case theresults of the abstract doctrine and of the experimentalpractice would coinc ide in every respect.”My object has been to assist in removing the incubus ofempiricism from artillery sc ience, and whilst fully consc iousof the imperfection of my efforts, and of the opportunity Ihave given to adverse criticism,I will only say to my critics,Si quid rectius novisti, candide imperti.”J. A. LONGRIDGE.CON T EN T S .CHAPTER I .EXPLOSIVE SUBSTANCES IN GENERAL.Definition ofExplosive SubstancesDistinction between Explosion and DetonationInstantaneous reaction impossiblePercussive and Static Action of GunpowderForce of Explosives ; Roux’s experimentsBerthelot’s two Classes of ExplosivesPotential of ExplosivesNitrate and other PowdersPhysical Characters of Gunpowder—Density, &c .CHAPTER II.FIRED GUNPOWDER.Liquid, Sol id, and Gaseous ProductsD issociationIgnitionCombustionEffect ofRate of Ignition and Combustion on PressureForm of Grain andRate ofEvolution ofGasSpherical GrainCubical GrainPrismatic GrainDisc , Mr.Qui ck’s PowderEffect of Formon PressureProducts of CombustionVolume of Gas and Units ofHeat evolvedTemperature of CombustionStrength of PowderEffect of Cooling by the Metal of the GunM . Sarrau’s RemarksTension of GasesForce of the PowderAction of Gases Expanding— Two HypothesesM . Sarrau’s Examination of the two HypothesesComparison with ExperimentMovement of Products of Combustion in a GunX CONTENTS.CHAPTER III.M . SARRAUgases may be calculated by the ordinary laws for gaseousmatter, deducting from the volume of the containing vesselthat of the non-gaseous matter at the temperature of combustion.Let thenC volume of containing vessel .10 weight of powder .To absolute temperature of combustion.a volume of non-gaseous matter arising from unitof weight of powder at temperature To.volume of the permanent gases arising from unitof weight of powder at temperature zero andatmospheric pressure.p , atmospheric pressure 103 °33 kilog. per squaredecimetre).p pressure after explosion.Then 0 a w vol. of permanent gases at To,50 INR ENAL BALLIS1 7CS.2 Tand” C - a w'm3p = f0 _( 1 )107. If the vessel be not entirely filled with powder, thedensity of charge (gravimetric density) must be taken into106 ,Qaccount. Let this be represented by A, then A( 1) becomes19 = f'1108 . The symbol f is what is called by M. Sarrau theforce of the powder, but it must not be confounded withthe absolute pressure or tension of powder exploded in ac lose vessel originally filled with powder, and which wasdetermined by Noble and Abe l to be about 43 tons persq . inch or 6568 ki log. per sq . centimetre .By force of the powder M. Sarrau denotes the pressure of the permanent gases arising from 1 kilog . of powderat the temperature To ofC ombustion when oc cupying uni ty,i. e. 1 dec imetre cube or 1 litre of space , that is to say,when exploded in a vessel whose capac ity is 1 a , abeing the volume occupied by the non -gaseous matter.The value of f p °21$3T°and for the same powder19° v, T° is assumed to be constant, and independent on the273gravimetric density of the charge .so is the volume of permanent gases at temperature zero, andin powders of the same composition may be considered asconstant. In order that To should be constant, it is necessary, first, that the exterior work be nil, which is of courseHere the unities are the kilogramme and decimetre, and since 1 kilog. ofwpowderoccupies 1 A 6https://www.forgottenbooks.com/join52 INTERNAL BALLISTI08 .potential is nearly 40 per cent. greater, whilst the actualeffect is nearly the same.11 1 . This last remark was confirmed by actual experiment by Messrs. Noble and Abel, who found the actualpressure fired in a close vessel to beWalthamAbbey powder, 43 tons per square inchM ining powder 44and from this they conclude that the capac ity for performingwork of the various descriptions of powder is not verydifferent. This conclusion is not borne out by the calculations, the results of which are given in the last table.It is true that cocoa powder was not in use at the time ofthese experiments, but excluding it, there appears to be adifference between Spanish pellet and mining powder ofabout 12 per cent. in favour of the latter,whilst the potential was 50 per cent. greater in the former.112. If the above results are even only approximate lytrue, the disadvantages of cocoa powder seem apparent:notonly is it a very weak powder, involving much heavi ercharges for equal ballistic effect, but it is a much moreexpensive powder andmore difficult to manufacture .In the evidence given by the Superintendent of the RoyalGun Factory before Lord Morley’s Committee in 1887 it wasstated that the costs were as followsPebble manufactured at WalthamAbbey 46 4bought from the trade 60 0Cocoa prism, brown , WalthamAbbey 85 7Westphalia Company 109 7}Rothw eill Company 1 10 5Chilworth Company 123 91 13. When it is borne in mind that the cocoa powder is24 per cent. less powerful than the pebble, it is apparenthow disadvantageous it must be in an economical point ofv iew. But the disadvantage does not st0p here . Theincreased bulk of the charges,involves larger chambers inthe guns, larger cartridges and larger magazines. Nor isthis all , the inc reased bulk of the products of combustion,INTERNAL BALLISTICS. 53and the increased temperature of combustion,must increasethe erosion of the bore , and it is probable that much of theerosion which is becoming so very serious in our modernartillery, is due to these causes .The presumed advantages of cocoa powder, viz. the lowpressures obtained, are due chiefly, if not entirely, to thesmall surface and slow rate of ignition, and the consequentdisplacement of the projectile before the whole charge is consumed,thus keeping down the maximum pressure in the gun.1 14. There is a difference of opinion among artillerists,with respect to the action of the products of combustionwhilst expanding in a gun .The first hypothesis, which is that of Messrs. Noble andAbel,is that the non -gaseous portion of the charge in aliquid state is very finely difi'used throughout the gases,andis always at their temperature, thus giving out heat whilstthey expand, and consequently to that extent preventing thefall of pressure.The second hypothesis, that of Messrs. Bunsen andSchischkofi',is that the temperature of the non-gaseousportions remains constant or nearly so, and thatwhatever heatit gives off is simply radiated to the walls of the chamber,without affec ting the temperature of the gases.l stHypothesis.115. In a lecture readby Captain Noble on 3rd April, 1884,at the Institution of Civil ’Engineers, on Heat Action ofExplosives,”he says, In the researches made by Sir F.Abel andmyself, when we found that the pressures in thebores of guns, and the energies generated by gunpowder,were far in excess of those deduced from Bunsen andSchischkofl's theory, we came to the conclusion that this differencewas due to the heat stored up in the solid, or rather theliquid products of combustion . In fact these products,forming as they do nearly three -fifths of the weight of thepowder,be ing also in a state of very minute division, constitute a source of heat of a very perfect character, and are54 INTERNAL BALLISTI08 .available for compensating the cooling effect due to theexpansion of the gases on the production of work.1 16 . Captain Noble does not state in what way the greatxcesses spoken of were determined, but it is to be presumedthat he is comparing the ac tual results obtained from theenergy imparted to the projectile , with the results whichwould be deduced from a pressure curve formed on Bunsenand Sch ischkofi’s theory. If so I am quite unable to accepthis conclusions. The energy as measured from pressurecurves, is always greater than the actual energy imparted tothe projectile , as various other resistances have to be overcome , as will be seen hereafter, and unless it can be shownthat these resistances are greater than the above differences,there is no need to seek for a source of heat in the nongaseous products.1 17. M. Sarrau has investigated this point by calculatingthe ballistic result,according to the two hypotheses in threedifferent guns, and comparing these with the actual resultsobtained by firing .The following is his method of procedurel st Hypothesis.W exterior work done .E mechanical equivalent of heat.0 mean spec ific heat of products of combustionat constant volume .3; F(t) weight of these products atany time t.T the ir absolute temperature at same time .T, initial absolute ”temperature of combustion .p , atmospheric pressure .p pressure of gases at time t.v, specific volume of gases fromunit of weight(1 kilog.) of powder at zero and atmospheric pressure .V volume of the gases at t.INTERNAL BALLISTI08 . 55T) ;Eliminating T between (1) and (2) and observing thatpaveTo po”0f273and wri ting 2 0for273 E cp v + 2 9 W = f y.mass of the projectile.area of bore or transverse section of projectile.distance moved by projectile at t.length of initial void, or length of bore whichwould give the same capacity behind theprojectile as the vacant space before theprojectilebegan tomove and consequentlyV is thus made up of(a) V., initial space in the chamber not filled withthe charge.(6) y, interstices between the grains of powderwhich are unburnt at t.(c) co l the cylindric volume generated by themotion of the projectile at time t.When the whole of the charge is burnt there should beadded, the original volume of the powder, less the volume ofthe residue of non-gaseous matter at the temperature ofcombustion, but as this is practically equal to the originalvolume of the powder, this item is reduced to zero.Consequently56 INTERNAL BALLISTI08 .but, as before,V w (lThereforeIand z == -aNow supposing the ignition and combustion to be instantaneous,i . e . before the projectile moves,V, y, is constantand therefore z is constant in this case .Calling so thisparticular value of z,its value islt,d’c 1 axaz+a: an a:(at),as 1+ z,'d t d t z+ z, d tz d iand replac ing c_i_as by its value fromd tdate a: (1 21fromwhich it appears that the law of the accelerations is thesame as that of the veloc ities.135. The density p varies throughout the mass accordingto an unknown law, but the variation is probably not great,64 INTERNAL BALLISTIOS.and may be neglected in the calculation of terms whosenumerical importance i s relative ly small .Under this hypothesist12 a:3p to2172"1+ 1, dz2p to d l 28“(1+ (cl— t) ”2 d”The integral as”d a:is to be taken within the limits(l l,)3l I, and 0and lts value 1s therefore3Substl tutlngwhich,and observing thatPw (I 10) [1 0we get15 P (l lo)therefore we getThis value is the superior limit, and as only about1eoths of the charge is reduced to gas, the real value may notexceed 1540‘136 . When the reduction to gas is total, the coefficient1 1°is reduced to unity .l 2When the reduc tion is partial andprogressiveBapproachINTERNAL BALLISTI08 . 65to unity with increasing values of l,and may still be considered as 1 without inconvenience .When I is very small,Bdifl'ers from unity, but in this casea and a , are very small, and the influence of these terms isnegligible .137. If in (23) a a , be made if we getdz i d i za sso that the essential form of (3) remains the same, therebeing a slight change in the second side only,and this maybe treated as an increase of the mass of the projectile byg.If fir the mass of the projectile may be considered asinc reased by one-ninth.M. Sarrau points out, that within the usual limits ofprac tice , the ratio adoes not vary much, and in such casethe effec t of its introduction in (27) is virtually to reducethe value off.In like manner,he says that the effect of cooling by transmission of heat to the metal of the gun,may be taken intoaccount in estimating the value off . This is no doubt true,and it can lead to ve ry small error, provided the value of fas estimated is applied to bal listic elements not too widelvdiffering from those of its estimation.138. It is very truly observedby M. Sarrau, that the a.p rioridetermination of f, by the methods above given, almostloses its practical value, inasmuch as we are not in a positionto apply to it the proper , reductions due to the various causesof loss o f work.On the othe r hand, too much importance must not beattac hed to this. He says, It would be very diflicult, ands66 INTERNAL BALLISTI08 .of no great practical utility to obtain a formula which wouldpermit the calculation a priori of all the effects, withoutborrowing from experiment certain practical data ; but atheory which by obtaining from a few practical experimentscertain coeflic ients, and which would then serve to calculatethe effects realisable in other conditions than those of theactual experiments, would evidently be of no great practicaluse . What is really required is to establish the form of therelations which bind together the effects obtained, to theBallistic Elements,” and thus to avoid the use of purelyemp irical formulae, which in the absence of any naturalguidance , may be found incompatible with the nature of thephenomena which they are intended to represent.139 . The formulae general ly given in treatises of Ballisticsare purely empirical they embody the laws which combinetogether the veloc ity of the projectile with certain ballisticelements, such as the weights of charges and projectiles, thelength and calibre of the guns, but they do not take accountin any way of the peculiar properties of the powderitself.It is here that M. Sarrau has stepped in and by theguidance of theoretical considerations, has succeeded in a veryremarkable degree in obtaining formulae for veloc ity andpressure , into which the spec ial character and nature of thepowder, its force , the rate of its combustion,and the form ofthe grain are all introduced, and thus the discussion of theseformulae has led to very important conclusions relatively tothe conditions of loading, and the nature of the powder suitable in every case.The following chapter is devoted to an explanation of themethods followed by M. Sarrau,and it is indeed little morethan an abstract of his own writings, which, by his kindliberality I am permitted to make use of.https://www.forgottenbooks.com/join68 INTERNAL BALLISTICS.feeble absorbing power, and, as has been previously shown,there is no defic iency in heat, when the actual results arecompared with calculation, to necessitate any such collateralsource of heat as is supposed by Noble andAbel.Diferential Equation of Motion of Proj ecti le.142. Let g weight of powder burnt at the end oftime t.a distance passedthrough by projectile at do.P, mean pressure of gases in kilog. per sq.metre .V, volume of the space behind the projectileless the original volume of the charge incubic metres.11 veloc ity of projectile.143. If at the time t, the charge ceased to burn, the gasesalready formed would continue to expand adiabatically,and if P be the mean pressure and V the volume at a timet1 greater than tPIV,"where n is the ratio of spec ific heat at constant pressure toC,’ 2324that at constant volume0. 17621 319Let W exterior work done at the time t, , then the workdone at dt1 will be d .W, andd W P dV.Introducing the value of P from (1) and integrating(3)but when t 131 , V V1 , andW 5m (neglecting thev is viva of the charge), thereforeC = §mv2+Noble andAbel, 2nd Memoir.INTERNAL BALLISTIOS.nIf now the gun be supposed indefinitely long, and Vtends to infinity,W tends toriOfIand 51 Vi V1"tends tozero, since n is greater than unity.Therefore144. ThatW tends tolOf g 18 thus show?l l . ‘y ( 4 1nm be the we ight of a charge burnt, the gaseswill occupy at the origin a volumetherefore P, V, P, 1:q.The product P, v is constant, whilst u varies, so long asthe temperature T, is constant, but when u 0°001 cubicmetre the pressure is what M. Sarrau calls the force of thepowder and denotes by fIf therefore f be expressed in kilogrammes per squarecentimetre, and P, in formula (4) in kilogrammes per squaremetreP, v 10,000f x°001 _ lOf ;or making v Y/UCl; P V Wh o/A d0 0:When the gun is prolonged indefinitely $l tends to zero,and therefore W tends toqii LIas stated in the precedingparagraph.INTERNAL BALL] 8 TI08 .145. Equation (5) may be transposed thusalsoSince v and V, m (w z) and P, w difl'ers veryd tdas“ dy.l ittle from the movmg force , 01 md 12 yInserting these values in (5)NR“ zaj zzn — l 7“or including the factor 10 in f, which is only changing theunity of measure off, which then becomes hectogrammes persquare centimetre ,“a,146 . This formula, which is the same as that arrived at byM . Sarrau, has been deduced by a somewhat shorter methodby Captain Roulin of the French Artillery , from whom theabove is borrowed.It is not rigorously exact because in deducing it severalelements ofminor importance have been neglected, but as weproceed to show, all these may be taken into account in thefactor f, which is a purely numerical factor ascertainable byexperiment.147. (a) The gases are to some extent cooled by contac twith the walls of the chamber, and therefore the work done”fibl by a quantity depending on this coolingac tion, and this quantity is one depending on the we ight ofpowder burnt as compared with the amount of coolingsurface . Were this determined, the error might be compensated by altering the value off .is less thanmy”2half the vis viva of the whole system,including the projectile,the charge, the gun itsel f, and the carriage . Consequentlythe value ofm is increased.but(b) At any time t, the work done is not strictlyINTERNALBALLISTICS. 71The real value of this term is in fac twhereM mass of gun and carriage .pt mass of the charge .p radius of gyration of projectile .R radius of the bore .0 angle of rifling.fv‘ veloc ity of recoil.The value of the factor within the brackets is generallynot much greater than unity, because generallytan 9 °005,° 1666and in any case these terms are independent of the time ,and consequently may be made to enter into the value ofmmu“2(0) Since there are passive resistances opposed to the“ascl—t73value, consequently this value of the second term is toosmall, and to make up for the passive resistances it is necessary to increase 771 in this term.147a. Consequently, in order to make equation (6) correctit is necessaryl st. To diminishf on account of cool ing.2nd. To increase m in the two terms of the 2nd member ,or what comes to nearly the same thing on account of themmotion of the projectile, P co is not m buthas a greater72 INTERNAL BALL] 8 TI08 .formof the equation, diminish still further the value of f inthe first member.Hence all the corrections relative to the neglectedelements may be included in the value ofj ; and this quantityis therefore simply a numerical coeffic ient, the value ofwhich must be found by experiment.148. The integration of (6) 11 ill give the form of the relation which unites the effects, viz. velocity and pressure, to thevarious and variable ballistic elements.Writing 0 Eat— 1equat1on (6) becomesdzw (l a:2(x z) 71720 ( ll— t.0being a constant, n was formerly estimated at butthe more recent estimate by Noble and Abe l is 1 ° 3l .l 48a . Before p roceeding further it is necessary to makea few remarks respecting the Combustion of the charge .149 . Le t i t) be the we ight of the charge and q the weightof the powder burnt at the time t, then g is a func tion of tand may be represented by we (t).«l» (t) depends on the form of grain andon the rate of burning of the powder in free air, and before proceeding with theintegration of (7) it is necessary to determine the form ofthis function .150. As was previously remarked, a distinction must bemade between Ignition and Combustion, and in every case , ifa high ballistic effect is sought for, ignition should be asnearly simultaneous as possible throughout the charge .With large -grainedpowders thi s is obtainedby the frequencyand size of the interstices between the grains, whilst withfine -grained powder 1t may be obtained by a hollow spacesuch as a perforated tube extending throughout the fulllength of the cartridge . With very large charges it mayhappen that the Ignition of the front part of the chargetakes place at so late a period that many of the grains leavethe muzzle of the gun long before they are consumed. It isneedless to say, that this is simply had artillery practice andINTERNAL BALLISTICS.it would it be folly to seek for any ballistic formula suitablefor such cases ; nor has this been attempted by M . Sarrau.He assumes that the period of Ignition ‘is very smallcompared with that of Combustion, an assumption which Ibelieve ought to be realised in all cases.151 . It is therefore with Combustion that we have now todeaLLet then S be the ignited surface, which is supposed to bethe whole surface of the grain ;V,the ve loc ity of combustion at any time t8,the absolute density of the powder ;n, the rate of emission of gasThe volume burnt in d i SV d t,andThe we ight d t SV 8 at t.Consequently the rate of emission 1 e . the volume evo lvedin cl tSzf‘“s v a.152 . It was found by Piobert’s experiments that thevelocity of burning varied inversely as the density, consequentlyV8 is constant, and the rate o f emission is proportionalto the surface simply,but this surface is constantly varyingand the variation depends on the form of the grain.153. The case of Combustion in free air will first be considered. The grains be ing homogeneous and spherical, andthe combustion being in free air,of course the pressure isconstant.Now the velocity of combustion andthe density being constant, the velocity of emission is simply as the surface , andthis surfac e is proportional to the square of the radius, amas the radius decreases uniformly, the emission of gas isinversely as the square of the time .4 rrR3IfR be the orlgl nal radius, the or1g1nal volume 18 3at the end of the time t, it is reduced to4T; (R V t)3andthe weight of powder burnt will beapn(rt - v i)?74 INTERNAL BALLISTIOS.If 1 be the total time of combustion of the grainT V = R or V I},1.t 3 t°3m il (1 l= w°i1 (1 zl iwhere w, is the original weight of the grain.The velocity of emission at t is-iVVhen the grains are not exactly spherical, nor ofexactly the same dimensions, the same formula may be used,taking the radius of the mean spherical grain to calculate7 . If for instance N be the number of grains in 1 kilogrammeof powder, R, the radius of the mean spherical graingw RfiSN l ,154. When powder is burnt in a gun, the same law ofemission would hold if the veloc ity of combustion wereuniform, but this is never the case in reality, since theveloc ity of combustion depends on the pressure. Now asthe evolving gases are confined by the projectile, and as theemission of gas is very rapid at first on account of the largesurface , it follows that the pressure rises rapidly at first,andthis again by increasing the ve loc ity of combustion,furtherinc reases the emission of gas, and the result is that beforethe projectile has moved far, a very high pressure isestablished, but as the projectile rapidly acquires veloc ityand increases the space behind it,the pressure begins to fall,and it falls rapidly, not only on account of the increasingspace but on account of the loss of temperature in the gasesdue to the conversion of heat into energy, and also to thedecrease of the veloc ity of emission due , both to the rapidlyhttps://www.forgottenbooks.com/joinINTERNAL BALLI8 TICS.area of OM B beyond the same ordinate , and therefore thepart 0N C must be less than 0MB.From this it is evident that the ao- called progressivepowders,even if their manufac ture were possible , must givea lower ballistic effect than ordinary powders, although ofcourse they strain the gun less ; but the same result may beaccomplished much more easily by attending to the formand dimensions of grain.156. When the size of grain is increased the initial surfaceof the charge and the rate of decrease of the veloc ity ofemission in free air are both dec reased.For let there be two charges of the same weight of cubicalgrain powder, and let the length of the side he a and thenumber of grains N in one charge, and the length of the sideN8Thus the surface of the first charge N x Ga”,and ofN x 6 a2the second x 6 x (2 a)i —2or just one-half of the surface of the first.Also if V be the veloc ity of combustion in free air, thenat the end of the time t, the dimensions o f the grains of thefirst charge will be a 2V t, and of the second 2 a 2V t, andthe total time of burning will be, for the first, t2—7andfor2 a in the other, then the number of grains will beathe second, t, V2 ‘tConsequently the veloc ity of emission of the secondcharge is less at first, and decreases more slowly than in thecase of the first charge with smaller grains.157. The same result of decreasing the velocity ofemission, might be attained by increasing the density, andas a matter of fact, the usual method of obtaining what arecalled slow powders is, both to increase the density and thesize of grains. It must, however, be remembered that indecreasing the velocity of combustion and thus obtainingslow powders, we increase the time of burning of the charge ,andmay do so to such an extent that a considerable portionINTERNAL BALLISTICS. 77of itmay be blown out unburnt. This of course necessitatesan increase in the length of the gun , with all its attendantpractical inconveniences.158. An approach to uniformity of emission so far asthat depends° on surface , is made in so-called disc powders,that is to say in powders pressed into thin discs of uniformthickness, and of the same diameter as the gun chamber,so assembled together as to permit simultaneous ignitionbetween adjacent discs. In this case the surface be ingnearly constant, the veloc ities of emission in free air wouldbe nearly un iformandthe time of total combustion dependenton the thickness of the discs.Mr. Quick has patented this description of powder andis apparently getting very excellent results in a 4- inch gun.The difficulty that has been met with by others is that ofthe breaking up of the discs during combustion, thus givingrise to irregularity of inflamed surface and consequent irregularity of emission of gas. It is probable that this practicaldifficulty will be overcome, and that disc powder will befound very advantageous in guns of large as well as of smallcalibre .159. In France an approach to the advantages of discpowder has for some years been obtained by the use ofprismatic grains in which the thickness is small relative tothe length and breadth.For instance, let the grain be of the form shown in thediagram, of the lineal dimensions a , B, and y , of which a. isthe smallest. In this case the time of total combustionTINTERNAL BALLISTI08 .If S 6 N a2 be the surface of the charge composed ofcubical grains whose sides a this charge wi ll be consumeda7 7°For the flat grains the original surface of the sameweight of charge isin the same time 7Sl = N xand making5 as and 2 y we getThe surface at the end of the time T will beSnl = N x 2 { (B— 2 V f ) (7 - 2 V f ) l ;or, since 1v2 SSt]. 2 k _ (B— a) (y x) ( l y) ,and since a:andy are each less than unity we have SI less thanS and S,Igreater than S ; therefore with the flat grains theveloc 1ty of emission is less at the beginning, and greater atthe end of the combustion,that is to say it decreases lessrapidly .It is therefore evidently advantageous to make the ratiosas and y as small as possible consistent with the grains notbeing broken up under the pressure of the gases in a gun.In France , with powders o f high density (about 1 °78 to1 ° 8) the values of asand y vary fromg to i , and if the valuea y g, be adopted, we find from the foregoingformulaSI 7 S, and SI]“917 Sr 01‘ S11INTERNAL BALLISTI08 . 79so that the advantage of the flat grain is very considerable .160. The weight of powder burnt at the end of the timet is obtained as follows. Let, as before , V be the veloc ity ofcombustion under pressure p , and let a B and 7 be the lengthof the sides of the grains.At the end of (i i these become2 t, B 2 t , 7 2 t ,and at the end of the time t- 2J’d , s - z t , r— 2J’t ,0 0the volume of the grain will be(a. 2Jo i)(s 2 t )(7 2"d ),and the volume of powder burnt will bea By v.Calling q» (t) the ratio of the powder burntthe time t, to the total weight of the charge,(1 37 — 0 0aBrand replacing u by its value found above(a) if “) 17161 . In free air V is constant, and if it is denoted by(b) 1t (t) = l 1If 1 be the total duration of combustion of the graina. 2 V0T 0,80 INTERNAL BALLISTI08 .whenceusing which in (b)1andmakingx _aB’r1 = 1which may be put into the formor if we writea + y + a ya1 + z + y,t t t2-72The weight of powder burntwill be w . «l» (t)andthe veloc ityof emission orWith cubical grains as y 1,andW1th flat gra1ns, 1faz= y = 3‘sA= l7 32 t 4 123_°i° 9 72 4-523INTERNAL BALLISTI08 .162. The formula (a) 160) may be applied to the combustion'in the chamber of a gun, only in this case V is nolonger constant, but is a function of the pressure .M. Sarrau arrived at the conclusion, that the relationmight be expressed by the formulaV ==Kp a',and that the value of a'was 5, therefore if V, be the veloc ityat atmospheric pressure p,V = 5.If therefore p be taken as the pressure in the chamber, thea bove formula becomeswhich, proceeding as before, may be put under the formA. a'z(a) 111 0) (J’d z{ 1 2 ) a r e; a,»In like manner for any other form of grain, the combustionunder pressure may be deduced from the combustion in theopen air by substituting for t the definite integral163. In the case of prismatic powder with a central holethe calculation is as followsLet p be the radius of the central hole ;B, themean radius of the inscribedand c ircumscribedc ircle, which for simplic ity is used instead of thereal periphery.h the height of the prisms.82 INTERNAL BALLISTICS.Then the original volume 71 h (32and at the endof the time t it will be11 11° { (R V,t)2(p (h 2 Vet),and the volume of powder burnt will bewasp2) h v,and the function7r (32p2) h v71'(R2p2) hUsually R p h,2Vor = R -p, or V,where r is the total time of burning, and substituting thisfor u,« (as -p2) (ifigh t)m e s h es- we);https://www.forgottenbooks.com/join84 INTERNAL BALLI8 TIOS.Nowmbeing themass of the projectile, w the area of the bore ,m dzm.”cl—IVd2 a:a ’32003) = (wmpoand making useoof this instead of t in 111 t we get9 w 4! (t) w; (32nd ,A m p. m 2 0.(M2 a!1° (w dt + (m) 51W“)Therefore combining (7 and (8) we getdz a(w z) d (3 (1:7 )wA a’2 0,2fi r —5150) 5 )W ]zindex 7 be suitably chosen .86 INTERNAL BALLISTICS.Making these substitutions (9) becomes(my ”ill -ii“ ?Replac ing z by its value 118)w w”A 81000 wand observing that m YV 7462where c is the calibre of9the gun, andW the weight of the projectile , we getf a 5 7v = 1 ( l x“ l’ il lA and B being numerical coefficients v iz .1000(pof f-vi“) and Bdiffers very l ittle from a} , for which valueiattains its maximum 4; therefore we may1A 4 A“introduc ing which into (11) we getA(W l)é" 2 7 8 7'i l1 _ B; c iIt has been shown by the experiments of Col . DesbordesMémorial de l ’Artillerie de la Marine,’ vol . vii. p. thatthe veloc ity varies as the %th power of the we ight of thecharge when the gravimetric density densité de chargement is constant, and as the i th power of the gravimetricdensity when the weight of charge is constant.Consequently we mustmake 7 g.The division 1000 is introduced to bring the unity of volume of z intocubic decimetres .https://www.forgottenbooks.com/join88 INTERNAL BALLISTICS.the subscriptmdenoting themaximum value of the functiond2 311dE2The function go is a purely numerical function ofE, and ofdz yodE”which may be denoted by N.Replac ing K andBby the ir values, and writing aP = Nf. (Zmso that the maximum value of this function is a number("Poand smcemw 1 110000, A aWriting8 N x 101?11 (9 110035and'since 85 is very nearly constant,including it in K weget finallyP KQA(w l)%02and sinceP= K a2 A (15)which is M. Sarrau’s formula for the maximumpressure onthe base of the projectile.Mcairn/umPressure on Breech of Gun .175. The above formula deduced from the acceleration ofthe projectile, gives the maximum pressure on its base .From it may be deduced the maximum pressure on thebreech as followsLet M,m, and P be the masses of the gunwith its carriage,the projectile, and the charge respectively, and c l , 0 theveloc ities of the two former,INTERNAL BALLISTICS. 89As regards the veloc ity of the charge , it is evidently lessthan that of the projectile , and it may be denoted by 61 0where 01 is a coefficient less than unity.Consequently mu M u, 6 1 mu O, and differentiatingwith respect to the time ,whenceand denoting by P0 and P the pressures per unit of surfaceon the breech and base of projectile respectively,p o = P (1 + oli f01 be taken 4.176 . The value of 61 g is based upon two hypotheses ofGeneral Piobert(a) That the density and temperature of the products areuniform throughout the space between the breech and theprojectile.(b) That if the whole mass be divided into infinitely thinslices at right angles to the axis, the veloc ity of each slice isproportionate to its distance from the breech.Neither of these hypotheses is exact. Moreover, inmaking P g ill—1;its value is certainly somewhat toosmall as there are certain small passive resistances whichhave been neglected, such as friction, &c . There is probably also a certain amount of vis viva lost, during theprocess of combustion, by the gases striking against thewalls of the chamber. Having regard to these and otherconsiderations, M. Sarrau concludes that the true value of01 is 3, and the above formula becomes P0 P (1which appears to agree very well with the results of1, [QJJ J U/ LQA )‘90 INTERNAL BALLISTI08 .experience. Consequently , to obtain the maximum pressureon the breech, the pressure on the base of the projectile mustbe multiplied by the factor 1 3 w2 Wand we getfor the breech pressure.177. For practical use, a monomial formula is moreconvenient, and this may be obtained by observing that anyincreasing function may, within c ertain limits, be consideredproportional to some positive power of its variable, therefore makingwe get(12‘7 J3 A(wWfi.W) 7'or making RI K Ko and 7 i , which has been foundsufficiently to agree with practical results, we get finallyK0A (w W)%02which is M. Sarrau’s formula for the maximum pressure onthe breech.Position of Proj ectile corresponding toMaaimumPressure.178. In 174) the symbol e was introducedl mz 03Now it is shown by M. Sarrau Mémorial de l’Artillerie de laMarine,’ vol. iv. p. 189) that e is generally very small, andhttps://www.forgottenbooks.com/join92 INTERNAL BALLI8 TI08 .and if aand equation (1@ becomesdzy (13102 C dz y go gig—z)181 . Although this equation is not directly integrable ,it is possible, as in manner following, to calculate the valuesof the function yo and its succ essive differential coeffic ientsfor progressively increasing values of the variables.For this purpose assumed2 90dcgand integrating“if,"m 1c“ c “ &c .and integrating againa b+ 2 + + 9+ &c .Introduc ing these values into the coeffic ients andexponents are determined, and series are obtained which arevery convergent, when g 1 or K3 t 1, that is to say, for1,or taking a'13this comes to Ké t 1 orlog‘ 1t or 95 ° 71 tof 186) orV = HH being a constant depending on the powder, and theexponents having the values given inThese exponents are six in number and are functions ofq and «y’.191 . If,in accordance with the experiments of Gavre, wemake a 3, we get from (25)Consequently B 2 «y 7’1} which agrees withthe empirical determination by the Commission de Gavre.Moreover, the relation 2 «y 7’2, gives- 2 % f t - 2 7 ’ flnd v = % —7 ;consequently the formula (26) becomesand it only remains to find 7 to determine the formulacomplete ly.192 . It has already been stated that the value of «7 increasesas the powder becomes slower. The values 5 and i wereH'INTERNAL BALLISTI cs.found, the first from a very quick powder, the second fromavery slow powder (i. e . relatively to the guns).The mean value 7 136may therefore be considered asapproximately true for usual conditions of fire, andadmittingthis value, the formula becomesV = Hand the value of the constant H, isH = M 6193. Under ordinary conditions 8 varies very little, sothat 8 5“ may be considered as constant, and, reduced to itsmean value, may be included in H,so thatE = M194. If 7! were taken 1 another formula, corresponding toa very slow powder, would be obtained, in which 0 does notappear, viz.V HIn this case 7’1 andH M sandfinally including 8 " 5 in MH M195. In general, formula (27) is applicable, and it has beenverified by M. Sarrau, by the actual results of firing withdifferent powders, and in difi'erent guns, and under difi'eringconditions offiring. To effect this verification it is necesseryhttps://www.forgottenbooks.com/joinINTERNAL BALLISTI08 .that is to say if the combustion of the charge were instantaneons.If, therefore, the above expression gives a maximum for afinite value of r,it is because the formula is only approximate. It was obtained by neglecting all the terms of a converging series after the first two. The value of r corresponding to those first two terms has, however, an importantsignification . It is a limit below which the variation of 7 hasonly an insensible influence on the velocity, and which it ist he re disadvantageous to exceed, because, whilst the veloc ityinc reases very slightly , the maximum pressure increasesrapidly in the inverse ratio of the time of combustion.Consequently, t he consideration of this particular valueof 7, called by M. Sarrau “the duration of the maximumdurée du maximum is of great importance in the presentquestion.198. Equating to zero the differential coeffic ient of V withrespect to 7 , obtained from equation and denoting by 7 1the value of 7 'corresponding to the maximumof V, we get3BA(W l)’6Here it may be observed that for a determinate form ofgrain, A is constant and the value of 7 1 depends only on thecalibre,the weight, and the travel of the projectile, and isindependent of the weight of charge and gravimetric density.199. When it is said, as it so often is, that a powderis slow or “ quick,” this expression does not reallydenote any quality in the powder itself. It depends chieflyon the conditions under which it is used. In fact, in agiven gun, the powder is slow,”when the duration of thecombustion of the grain is notably superior to that durationwhich, in that particular gun, corresponds to the theoreticmaximum of veloc ity, and rice uersd. Moreover, twopowders, fired in different guns, should be considered of thesame vivacity, when their durations of combustion areINTERNAL BALLISTI08 . 101proportional to the durations of the maximum, relativelyto the two guns used.200. Let, then, the ratio of the duration of the maximum relating to a given gun, to the duration of combustionof a powder in thatgun be called the Modulus of Vivacity,”or simply the Modulus,” and be denoted by as; then7.From this point of view, M. Sarrau adopts the followingscale of classification of powdersmodulus to. powder.201 . Accordingly, from since a:5(W etw= 3133Formulafor Initial Velocity as a Function of the Modulus.201 . Introduc ing a:in the place of r in formula (13) forthe veloc ity, a new expression is obtained which will be“(WW(5 198) (13)mayfound of great use . Since 7 ; 3Bcbe written thus102 INTERNAL BALLISTI08 .which making ait A iV i A (W?) (3and if 7 1 be replaced by its value and we takef tw) M 3 w)the formula for the velocity becomesf (x) . (37)MaaimumPressure on Base of Projectile as a Function of theModulus.202. The maximum pressure on the base of the projectileas a function of the modulus is thus obtainedExpression (15) above may be writtenP = Kf a A (wm) §c3andmaking as and {replacing r , in the denominator byits value 3BAw5P K sB“ 1 f-f’ 38A s203. The maximum pressure on the breech is deducedfrom this by writing Po for P and replac ing K by K0(Tani;thereforeP. a mamhttps://www.forgottenbooks.com/join104 INTERNAL BALLISTI08 .we get finallyiw; AI ci—nWt206 . This formula varies with n, that is to say, with themodulus as to which n is related by (40) or n if é 2It may be used, approximately, by attaching to n a constant value, in conditions of loading such that the modulusremains within certain limits.207. Among the different forms which the monomialformula for the veloc ity may take , those deserve spec ialattention which correspond to the value of the modulus{ PT and 155 .In the former case n i and in the latter n and theformula becomes when as g,t l i 13V Mw Ac,17;Wand when asV Mwhich expressions agree with those previously obtained208 . The former of these expressions is applicable towhat are called quick, and the latter to slow powders in thescale209. Since the velocity increases continually as 7 decreases, the value n , which gives a maximum, ought to beconsidered as the superior limit for the use of the Binomialformula.Consequently, this formula should not be used for powdersquicker than the powder of the maximum. This takesINTERNAL BALLISTICS. 105B30121)isgreater than the value 4 which corresponds to the maximum.It is practically advantageous to limit the use of theformula to cases where the modulus is below a less limitthan unity. For higher values, 7 becoming nearly equal toT 1, the theoretical expression becomes too rapidly stationary,and ceases to represent exactly the real variation of theveloc ity.If { it be adopted as the superior limit of the modulus,the binomial formula should cease to be used when the(“ll)’is found to be greaterthan a}, of 1913 or greater than 3.210. When the modulus is greate r than { if the monomialformula is applicable .In fact this formula agrees sensibly with the other, forvalues of themodulus approaching fr , and increases graduallywith the modulus, instead of passing through a maximum.It may , therefore, represent exactly the veloc ity, in all caseswhere the powders used act as quick powders, and this hasbeen verified experimentally by M. Sarrau, by comparing thecalculated velocities with those actually obtained under veryvarying conditions of loading.211 . Making use of the characteristics a and B theformula (42) may be writtenplace when the second term of the function 1value of the second term BBs wI A’ci l‘"1V=” M aBWWand to obtain the value of M,it is suffic ient to observe theveloc ity given by a powder of which the characteristics areknown, under given conditions offire.106 INTERNAL BALLISTI08 .Table of the Functionf (as).212. The function f (as) which serves to express the relation of the velocity as a function of the modulus, is reprosented by f (as) as} (3 as) when as is less than fir,and by Nmi when as is greater than Tar.The constant N is determined by equating these twoexpressions, making n fr .The following table gives the value off(as) for mcreasmgvalues of as from0°5 to 1 ° 2.213.Logf (0)1 °0352214. From (37) it is seen that when the duration of combustion of a powder is altered, all other ballistic elementsremaining unchanged, the veloc ity varies directly as f (as),and from (38) and (39) under like circumstances, thepressure on the breech and on the projectile varies as themodulus itself.Consequently the preceding table afi'ords the means ofcomparing the corresponding values of the pressure andveloc ity ; and it shows, that the increase of veloc ity is verysmall compared with the increase of pressure.For instance, comparing a powder of modulus 0° 6 withanother of modulus 1 2, the pressure is doubled whilst theveloc ity is only increased by about 5th part.https://www.forgottenbooks.com/join108 INTERNAL BALLI8 TI08 .under consideration corresponds to the maximum veloc ity ,we have x where r , is independent of w and A thereror dxMaking use of which in (44) and andmaking as beforeas xMm) 205)we getdw dAl8 + 1Adw dA drTo A 1Now by 207) n when the modulus is f r, and itis given by (40) when it is less than fr .It increases when themodulusdecreases belowfir, it is equalto 1 when the modulus rig corresponding to a very slowpowder.Variation of Velocity corresponding to a Constant Valueof MaximumPressure.218. By means of the above equations (47) and (48) wemay examine how the velocity varies, by the variation of theweight of charge, gravimetric density, and time of combustion, whilst at the same time the maximum pressureremains unchanged.219. (a) Let the weight of charge be constant, gravimetricdensity and time of combustion variable.l N'"71 Cu t l i t tc u t e» 7 u nwe t w“s ed; A k ic k L C; a] dL LQ/L Lx xv109éu V (bu /(Ev U u L”u { AA /b’l‘/ L ;Since (48) P0 18 constant -T)! 0,alsodw = 0,therefore(i f dAT Asubstituting which m (47)V i T_ 'n —A (i — n)If the modulus rc n 1 and i n is positive, therefore the veloc ity increases with the gravimetric density.From which the following proposition is derived.When the weight of charge remains the same, and thegravimetric density and the time of combustion increase,sothat the maximum pressure remains unaltered, the velocity isincreased, and the more so as the modulus of the powder isgreater, or the p owder quicker .From which it follows, that by using a very quick powder,and at the same time decreasing its gravimetric density,theveloc ity may be increased without increasing the pressure.220. (b) Let gravimetric density A be constant, and theweight of charge, and time of combustion variable.Since dP, and dA in (47 and (48) are each 0dV dwV ss — uwNow from (40) it .is seen that n y for x 0, and n hfor any other value of x,consequently Qn is alwayspositive , fromwhich is deduced the following propositionWhen the gravimetric density is constant, and the weight ofcharge, and time of combustion increase so as to keep the maximumpressure constant, the velocity is increased.221. (c) Let 7 be constant, w and A variable.Then dPo and d r O,and8VVasTherefore , When, with the same powder, the weight of charge110 INTERNAL BALLISTI08 .222. (d) When the capac ity of the chamber constant,andw and r variable.Let S capacity of chamber, then13 01A dwS A wtherefore (47) and (48) becomedVVd P0PoIt now w and r vary so that Po remains constant, we getfrom whicha"(fir (51)Now in ordinary conditions of practice n 19; so that Vincreases with 10, thereforeIn a given gun when the charge and duration of combustionincrease so that the pressure remains constant, the velocity isincreased.Consequently, with a size of chamber suffic iently large , theveloc ity may be increased without'altering the pressure byincreasing the charge of a powder for which 1° is greater.223. (e) From (50) we may determine the variation ofvelocity corresponding to a small variation of the durationof combustion.Suppose w remains constant, thendV drV rn increases as the modulus decreases, therefore , the samerelative variation of the duration of combustion has aninfluence on the velocity, the greater as the time of combustion is less.https://www.forgottenbooks.com/join1 12 INTERNAL BALLISTI08 .which the modulus is greater than unity, but M. Sarrauconsiders such conditions unfavourable in general. It is,however,obvious that this is a question which dependschiefly on the strength of the gun, and therefore by increasing this, higher ballistic efl'ects will be obtained withquick than with slow powders.227. In the reception of powder in France a certainmargin is allowed which is called tolerance .”In the manufacture of powder some irregularities areunavoidable, so that difl’erent lots of the same powder givedifferent veloc ities at proof. The limit of these veloc ities isfixed and is designated by the term tolerance .”By the foregoing formulae, the influence which thistolerance exerc ises on the veloc ity may be estimated.Suppose for example that the irregularity is due to avariation in the duration of combustion, and let 7 be theduration of combustion , which for a given form of grain,gives the normal velocity at proof, that is to say the meanof a great number of fires, and suppose that with a particularlot of powder, this duration receives a variation of dr , thenthe corresponding variation of veloc ity in any gun is givenby formulaLet no be the value of n in the éprouvette and V0 themean veloc ity of reception, thendV,Voconsequently,dV n dV,v n,’ v,Suppose then that dV, represents the maximumdeviationallowed at reception, the relation (55) gives the differenceof veloc ity which results from it in another gun. If then Godenotes the difference of the limits of reception,and e themaximum difference of veloc ities in any gun, we have5 :INTERNAL BALLISTI08 . 1 13It is to be remembered that n is expressed in function ofthe modulus according to and that when the modulus{ if we must according to (42) take n228 . Suppose for example , that the French powder Wii,is rece ived for the gun of 24mm. inches) with the conditions followingo 441 ms 50 9m. noIf the same powder be used in agun of 10mm. 4 inches)W 12 kilog., l 226 dm.,to obtain a veloc ity of 485 m.We haveV 485 as ntherefore by (56) we find6 20” ° 2which is the difference from the veloc ity 485 due to theirregularity which in the 24 mm. gun only gave a deviationof 9 m. from 441 metres.On the Constants contained in the Equations for Velocity and229. The equation for veloc ity isv A a (w l‘) (vis-J{1If then the value of the characteristics a and B are knownfor any particular powder, the constants A and B are easilyobtained by firing two rounds,with the same powder, butwith a variation in the ballistic elements,for in this way twoequations would be obtained containing the two unknownquantities.230. Now it is evident from the form of the above equation that we may assume arbitrarily values of a and B fromany one powder which ‘may be called the Type powder,”and for any other powder may find the relative values of114 INTERNAL BALLISTICS.a and B. This is what M . Sarrau has done, and he haschosen for the Type powder the powder known in France asthat is to say a Wetteren powder of which the thickness is 10mm. and the sides of the bases 13 and 16 mm.respectively. The density 8 1 °794, and the number ofgrains to the kilogramme or N 330 to 385, and for thispowder he assumes the valuesf 1, 7 1 . The composition of this powder isSaltpetre 75 0Sulphur 12 5Charcoal 12 5231 . As will be seen hereafter, the values of a and A forthis powder are a 2 A. O“851 , values depending entirely on the rain. 73 31And by Table 319)log 0. 0 20513log 3 1 -92993232 . Making use ofthese values and of the mean observedveloc ities obtained with this powder under the service conditions of firing in a 10cm. (4 inch) and 19 cm. inch)gun, in the equation for the veloc ity (13) the values of Aand B were found to belog A 3-16767log B 2 -18373The following are the ballistic elements from which theabove values were determined.Nature of gun .19 cm. gun 0 87010cm. gunUnities, kilogrammes and decimetres.The value of r I assumed, is that of the actual time ofcombustion of one grain of W113at the veloc ity of burningof 10mm. per second.https://www.forgottenbooks.com/join116 INTERNAL BALLISTI08 .Comparing which with the general formulas)Cubical Grain.235. Here as in the first place a B «y, and thereforea 3 A 1 ;236. These may be considered as spherical grains of whichthe mean radius is found as fol lows.N number of grains per kilogramme .8 absolute density of powder.R mean radius of grain.gwR°8N = 1 or Rand consequently as abovea 3 ; A 1 ;237 Flat Grain, Dimensions a, B, and 7 , of whicha is the least.Suppose thenINTERNAL BALLISTI08 . 117substitutingwhich values we geta° 8502 ; °2419.Cylindrical Grain with RoundHole.238. Let R external radius of grain.r radius of hole.h height of grain.Then, original volume1r (B.2r“) h.Volume burnt at t—rr (R2 — r‘) h — 1r (R2 2) h (1and ifR r be less than h2 v I (R — r)” t orR — r h r hand writingvolume burnt at t” (Rs so}. rr (R'r’) h (1 2)(1 x51),and the ratio of this to the original volume ist t t“2X1 “9a = l + w ;239. This may be treated as a cylindrical grain, usinginstead of R, the mean radius of the inscribed and circumscribed circles of the hexagon.1 18 INTERNAL BALLISTICS.On the Re lation between the Duration of Burning of aPowder and its Physical Prop ert ies.240. If e and 1 be the thickness and time of burning of agrain of powder, and v the corresponding rate of burn ingand since according to Piobert’s experiments the veloc ity isinversely as the absolute density , we have for grains differingonly in thickness and density7it being a constant.241 . M. Sarrau, adopting this relation, has applied it tothe calculation of the Characteristics of various powders,but a comparison of the values of 7 thus obtained,with thoseobtained by actual experiments, shows that the above formuladoes not exactly represent the law according to which thetime of burning depends on the thickness and density.In fac t,in the case of two powders W§% and of thesame composition and nearly the same density, the ratio ofthe thickness be ing that of the time of combustion is1 ° 25,and for two other powders SP3 and SF2 , of the samecomposition , the ratio of the thickness was 1 ° 84,whilst thatof the time of combustion was 1 ° 53.Again, if the duration of combustion was exactly proportional to the density, the veloc ities given by different powderswould be inversely as the 44 th power of the density,butexperience shows that the variation of the velocity isconsiderably greater than would be given by this law.Consequently, we are led to the Opinion that the actualduration of combustion increases more rapidly than thedensity, and less rapidly than the thickness.242. It is easy to perceive that this may be due in greatmeasure to the process of manufacture . In some cases, theamount of compression may be such as to ensure a nearapproach to uniformity of density”throughout the cakehttps://www.forgottenbooks.com/join120 INTERNAL .BALLISTICS.246 . The differences are inconsiderable except for theWH} and Cl , and are accounted for by M. Sarrau as due tosome difference in the mode of manufacture . The formulatherefore gives very approximate results, but should beapplied to other powders under reserve, in the absence offurther verification.247 . Although the formula 1:8 c is acknowledged to beinexact, it has nevertheless given satisfactory results whencompared with those actually obtained by firing.It is however preferable, in determining 7 from thephysical properties of a powder, to make use of thetaking the value of x from the table given above ; but aseven this formula is uncertain owing to the uncertainty ofthe law which connects x with the composition and mode ofmanufacture , it is always best to deduce the value of 7 fromballistic results actually observed.f or this it suffices to measure the velocity obtained undercertain ballistic conditions.For instance, the value of H is obtained from themonomial relationHwi AI l l!‘ c%and then 7 is obtained fromH = Mor denoting by the suflix othe data relative to the typepowder WH, the relation ofTis given by the relations(3(it-“fINTERNAL BALLISTICS. 121248. In this latter formula, replac ing the factors of thetype powder W{ % by the ir numerical values, we get7 N a‘ h 3 11- 8whencelog N 27 95399.It must however be borne in mind, that the above formulafor V ceases to be applicable when the powder ac ts in theéprouvette as a slow powder ; in which case recourse mustbe had to the binomial formula for the velocity .The Determination of the Characteristics of a Powder .249. The characteristics of a powder are represented byf a "m e)It has already been shown that for powders of approximately the same composition the value of f does not varymuch, and therefore if any one powder be selected as a typepowder, we may for that powder make f equal unity, itsactual numerical value being included in the constantsA andM.For the type powder chosen by M. Sarrau ,the thickness of the grain is 10 mm.,and taking the veloc ity ofcombustion as determined by Piobert to be in free air 10mm.per second, we get 1 1 .Now a and 7» are determined by the form of grain asshown above (5250. Consequently, for the type powder, we have f 1 ,7' = l,a Q7H? ° 850itiFor any other owdeg2"table1value of 7 may be obtainedthus, making f =(é) «a» (a? {1 320—2s122 INTERNAL BALLISTICS.V MGw i Ai ci lsecond of these is easily solved for 7 ; writingM wt Altat atV p ‘lfwe get7 :The first equation is not directly soluble, but it may beput under the formX Ywherex A al(w l) (WA)and from this 7 may be obtained by approximation .252. But we do not know a priori , which of the aboveequations is applicable when we fire a powder whosecharacteristics are unknown , in a given gun. To obviate thisdifficulty the following methodmay be used.253. The monomial formula is applicable when 17BE(251is greater than 2 73 ; if 7 2 73 the bi7'nomial formula is to be used.Moreover the two formulae will give nearly the sameresults in the vicinity of conditions which make 7 °273.This be ing so,apply first the formulaa‘ X87°A3and with the value of 7' thus obtained find the value of 17.If this value be the monomial formula is actuallyapplicable , and the value of 7 thus obtained is to beadmitted.https://www.forgottenbooks.com/join124 INTERNAL BALLISTICS.App lication of Formula to the Designing of Gun s.PROBLEM I.255 . Given the calibre and weight of projectile , to determine the conditions to be adopted to realise a given initialveloc ity and a given maximum pressure .256. The maximum pressure allowable is fixed by theresistance of the gun, and is the pressure at the breech . It istherefore this pressure which must be introduced into theformula.On account, however, of the greater simplic ity of the formula obtained, M. Sarrau resolves the problem with regardto the maximum pressure on the base of the projectile, andthen transforms the results, so as to introduce the maximumpressure on the breech .257. The calibre and weight of projectile being given ,the variables disposable to obtain the internal veloc ity V2andmaximum pressure on projectile P, are l, w, A, f, a,A,r ,the first three of which relate to the gun, and the last threeto the powder.If thenf is determined by the mode of fabrication of thepowder, a andAby the form of grain, the number of variablesis reduced to four, l, w,A, andNowi ll - BW w)c9and since V and P are given we have two equations forresolving the problem.258. If two of the variables w, l, A, 7 be assumed,the othertwo may be determined so as that the veloc ity and pressuremay have the required values.259. The two equations above , are only soluble when l andr are assumed or known, and w and A are the unknownquantities.v Aefu (aP = K& A (INTERNA} , BALLISTICS. 125It is, however, possible in all cases to put the unknownquantities under an explic it form by taking the modulus ofthe powder as an auxiliary variable.For this, it is necessary to consider the relations (37 and(38) which give V andP in function of the modulus a, and thevariables l, w,A, and add the relation (35) which existsbetween the modulus and the variables l and 7 .Thus there are three equationsas“,s H a,*(f)w Awff “Pm= 3Bwhich give the solution of the problem.Having given the modulus andone of thefowr variablesl, w,A, and r , to determine the other three, so that the initialvelocity andmaaimumpressure have the requi red values.Practically, the gravimetric density A is fixed withinnarrow limits, it is sufficient then to consider this variable asgiven. In consequence the problem finally resolves itselfinto this261. Having given themodulus and the gravimetric density ofthe charge, to determine the weight of the charge, the length oftravel of the proj ectile, andthe time of combustion of the grain,so that the initial velocity and maaimu/mpressure may haveIt is to be remarked that in the above the modulusappears as given with an arbitrary value. It may, therefore,be chosen a priori , so that its value shall be within suitablelimits as mentioned in262. The problembeing thus fixed as above, its solution iseas ily obtained from andThe two first equations give w and l. To eliminate l itsuffices to multiply together (37) and (38) raisedrespectively126 INTERNAL BALLISTICS.to the powers 2 and The value of w is thus obtained.From (38) we then get the value of l, and finally 7 fromThus,7 : H3where11 1 gal-asnyirx- lH, gA- 2(3B)- 5K%113 3 3was)- 2u se) f (w)"The values ofA,B, and K have been previously given .log A 3 ° 16767log B 2 ° 18373log Kfrom which are derivedlog H 1 10-02713log H2 o ~ 62937log H , 2 -66085263. Suppose now that instead of P the maximumpressureon the base of the projectile, Po the maximum pressure on thebreech is given.In this case, the formulae are derived from the precedingby replac ing P by P0, and K by K0 (fig-[Yand we gethttps://www.forgottenbooks.com/joinl 28 INTERNAL BALLISTICS.267. The expressions and (59) give the valuesofw, l, and 7'as functions of V,P,a, andA ; it remains to determine the laws uniting the unknown with the known quantities. These laws are very simple when the maximumpressure on the breech is amongst the latter ; they are somewhat less so when it is the maximum pressure on the breechwhich is given ; but in both cases the general drift of theformula is the same,andwe may therefore confine ourselvesto the first case in order to study the separate influence ofeach variable.For this purpose , in the expressions and (59)substituting the value of l in the equation for r , andwriting3A" l(3B) Kxc ) w h en)we obtainw vertw = Hl (-‘tiA2I : H, (f a tW_ _Y_A02 Pf H (QC“h wy —X(a) . (70)02P0268. Hence it appears, that in a gun of a given calibre,for given values of the maximum pressure, modulus, andgravimetric density,(a) The weight of charge, and the length of travel of theproj ectile are proportional to the vis viva of the proj ectile.That in a gun of given calibre and for fixed values of theweight of projectile, modulus, and gravimetric density,(b) The weight of charge is proportional to the squareroot of the maaimwmpresswre.The length of travel is inversely as the 3 power of themaaimumpressure.The duration of combustion of a grain is inversely asthe fith power of the pressure.INTERNAL BALLI8 TI08 .269 . From this it follows, l stly, that an increase in thestrength of thegun, permitting a higher maximum pressure,enables us, with a powder of the same modulus, and the samegravimetric density, to obtain the same veloc ity, by diminishing the length of the gun, increasing the weight of charge ,and using a quicker powder.270. 2ndly, That in a gun of given calibre, and with fixedvalues of the weight ofprojectile, veloc ity,maximumpressure ,andmodulus,The weight of charge is inversely as the gravimetricdensity.The length of travel is directly as the gravimetricdensity.The duration of combustion of a grain is as the squareroot of the gravimetric density.Consequently, by enlarging the powder chamber, wecan realise , with the same modulus, the same ballistic effect,by decreasing the length of travel, increasing the charge, andusing a quicker powder.272 . When all the ballistic elements remain constantexcept the modulus, the weight of the charge , the ‘length oftrave l, and the duration of combustion vary directly as thefunctions gb (a) «If (a) andx(a) the variation o fwhich is shownin the following table .273.274. Now,since the we ight of the charge is as (a) andthe travel as «p (a), and since a(a) increases andX(a) dec reasesas the modulus decreases,it is evident that the same ballistic130 INTERNAL BALLI8 TI08 .effect may be obtained with a slower powder, that is to saya powder of a lower modulus,by a simultaneous increase ofthe weight of charge , and decrease of the length of travel .Influence of the Nature of the Powder and the Formof Grain .275. The we ight of charge and travel of the projectiledepend on the nature of the powder and form of the grain,that 13 to say, on the factorf afax,and the length of travel directly as the square root of thef athe same ballistic result may be obtained with a smallercharge and increased length of travel, employing at the sametime a slower powder.The weight of the charge is inversely as the 3power ofsame. Consequently, by whatever means can be increased,276 . The value of fa7“may be increased either by adoptinga powder of different composition, such as the picrates, or bythe use of forms of grain giving a higher value to a, and alower value to 7x, such as a flat or pierced cylindric grain.Having given two difi’erent guns, to find the relations necessary between the weight of charge, the length of travel,and the gravimetric density, in the two guns, so as toobtain with given moduli , the same velocity andmaximumpressure with the same powder .277. Let Po be the maximum pressure on the breech, andV the veloc ity. Then Po andVmust be the same in the twogunshttps://www.forgottenbooks.com/join132 INTERNAL BALLISTICS.From(73)(W'of (W atc'a’c anfromwhich we get the values ofwhich relations give the solution of the problem.278 . The equation (79) gives1 W1 W'gits value from (78) and taking account ofthe relation (77) we haveat);but by (61) up (a) a”d) (a), thereforeand giving toand (81) becomesCombining (77 and the following formulae areobtained which give the solution of the problem.Ifc' W c'lEINTERNAL BALLISTI08 . 133279. If the two'guns are similar and theabove relations become280. M. Sarrau proceeds to apply these formulas to thefollowing problems1 . Calculation of initial veloc ity and maximum pressurein a given gun, under given conditions of firing, with apowder of which the Characteristics are known.2 . Determination of the Characteristics of a powder.3. Analysis’S FORMULA] .M . Sarrau’s Investigations of FormulaeD ifferential Equation for Motion of ProjectilesIgnition and Combustion of a GrainCombustion in Free AirCombustion in a GunEffect of Size of GrainMr .Quick’s D isc PowderSarrau’s Coefic ients a, A, p.Combustion of Powder as a Function of the TimeGeneral Equation of Motion of a ProjectileBinomial Formul a for Initial Veloc ityCharacteristics of a Powder (is? and3Formula for MaximumPressure on base of ProjectileMaximumPressure on BreechPosition of Projectile corresponding to MaximumPressureM onomial Formula for Initial VelocityTheoretical Maximum of Veloc ity7 , value of, for MaximumModulus of Vivac ity of a PowderFormula for Initial Veloc ity as a Function of the ModulusMaximumPressure as a Function of the ModulusLimitation of the Binomial Formula for VelocityTable off (at)Influence of Ballistic Elements on Velocity andVariation of Veloc ity , corresponding to a constant value ofmaximumpressureA constant, w and 'r variable—r constant, w and A variableCapac ity of Chamber constant, w and 'r variableComparative Variation of Velocity and MaximumPressureLimi ting Value ofModulusMargin of Reception of Powder tolerance —in FranceConstants a andBin Sarrau’s FormulaValue of A and BDetermination of a and ARelation between the Time ofBurning and Physical PropertiesDetermination of Characteristics a and BAApplication of Formula. ProblemI .Given calibre and weight of projectile, to find the otherconditions to realise a given velocity and maximumpressureCONTENTS. xiPARA. PAGEGiven the modulus and one of the variables 1, w, A,-r. todetermine the other threeGiven the modulus and A, to determine w, l, and 7Table of functions of ModulusWith a given modul us, to find the relations be tween to, l, and Aso as to obtain the same velocity and maximumpressure intwo different gunsNumericalApplication to ProblemI In itial Velocity andMaximumPressureDitto to ProblemII To determine the characteristics of agiven powderD itto to ProblemIII Theanalysis of a given gunDitto to ProblemIV. Given the calibre and weight of proj ecti le, ini tial veloc ity andmaximum pressure, to findtheinterior dimensions of gun and the conditions of loadingDitto to Problem V . ,Given weight of projecti le, charac teristics of powder, initial velocity and maximumpressure, to find the interior dimensions and conditionsof loadingTable of Characteristics of French Powde1sDescription of French PowdersD imensions of French GunsModuli of French PowdersFunctions of the Modul iAppl ication to Engl ish UnitiesCharac teristic of English PowdersTable of dittoPrinciple of Similitude of GunsCHAPTER IV.INTERNAL BALLISTICS IN RELATION TO GUNCONSTRUCTION.Designing of GunsAdvantage of the use ofWireResistance to Bursting StrainPressure CurvesNoble andAbel ’s CurveCurve of Initial Pressure, Mayewski’s InvestigationsCaptain A. Noble s investigationsPoint ofMaximumPressureApplication of Curve to determine Thickness of Chase of GunPressure Curve fromSarrau ’s FormulaComparison with Noble and Abel’s Curvexii CONTENTS.Longitudinal Strains behind Trunn ionsin front of TrunnionsAccident to Coll ingwood Gunto Longridge 6-inch Wire GunExamination of assumed causeStrain due to Inertia and Friction of ProjectileStrain due to Friction of Products of CombustionChamberingRiflingErosionCHAPTER V.GUNS CONSIDERED AS THERMODYNAMIC MACHINES.Count St. Robert’s viewsGeneral form of EquationDetermination of A Hn A VApplication of formula to 10-inch Woolwich GunSummary of resultsPercentage of useful effectGeneral remarksCHAPTER VI.CONCLUDING REMARKS.Reduction of pressure and increase of chargeNot necessary as regards strength of gunsNew PowdersStability of ConstitutionPOSTSCRIPT .American PowdersINTERNAL BALLISTICS.CHAPTER I.ON EXPLOSIVE SUBSTANCES IN GENERAL.1 . By the term 'explosive substance is meant a substancecomposed of two or more elements mixed together or chemically united,and such as, that when this affinity is disturbed,a violent reac tion takes place,giving rise to a great development of heat,and to various new compounds in a liquid orgaseous form.In the latter case their gaseous products expanded by thedeveloped heat constitute a reservoir of energy which isapplicable to mechanical uses.2 . The reaction varies in rapidity according to the natureof the compound substances.In some cases, such as fulminates, nitroglycerine , &c ., thereaction is extremely rapid, and is called detonation.”Inothers,such as ordinary gunpowder, it is much less rapid, andis called explosion ; whilst in others, such as fuse or rocketcomposition , it is still slower, and is called combustion.This distinction of terms, for what is in truth only onephenomenon, is at once unsc ientific and misleading, and ithas given rise to erroneous conceptions of the action of gunpowder, which wil l presently be considered.3. If by detonation ”it be said that, an instantaneousreaction is meant, the reply is, that no such thing as anR2 INTERNAL BALLISTICS.instantaneous reaction exists. Detonation, so called,isonly a very rapid “explosion,and “explosion ”is only avery rapid “combustion .”The most rapid decomposition which takes place withfulminates or nitroglycerine, is a gradual process, and theattendant rise of pressure in a c lose vessel, owing to theevolution of heat and gas, is a gradual , and not an instantaneous rise . The phenomenon is one and the same,theonly difference being in degree, and not in kind, and therefore it is desirable to designate it by one term only, that ofcombustion .4. The distinction of terms has given rise to curiouslyerroneous ideas as to the ac tion of gunpowder in a gun .We hear of the distinction between the percussive ”effectand the static effect of a charge of powder.For instance Mr. Lynal Thomas says, So far from theaction of the fired powder in a gun being that of a constantpressure estimated at so many atmosphe res, it is violentlypercussive and variable to an indefinite extent, the percussive action starting the projectile with a finite veloc ity,and he adds, that a distinguished mathematic ian whowitnessed some of his experiments drew up a formula inaccordance with the “ percussive ”theory, which formula hegives as follows02 = V2+V be ing the veloc ity with which the shot begins to more.It is hardly necessary to insist on the absurdity of a body11696a to move with a finite velocity.Again ,Colonel Bope Tspeaks of the percussive effect ofa charge of powder,” and the static equivalent of percussiveforce.”Action of Fired Gunpowder, by Lynal Thomas, Il lustratedNaval andil itary Magazine,’vol . i., 1884.t A Revolution in the Science of Gunnery,’read at the R.U.S. Inst ,23rd July. 1884.https://www.forgottenbooks.com/join4 INTERNAL BALLISTICS.very small, for the reason that the volume of gas producedand the units of heat generated are both small.6. In order to determine the relative force of explosives,M. Roux made a series of experiments by firing a givenquantity in an absolutely closed space within a large blockof lead,and he estimated the relative force of the explosionby the size of the cavity formed by the explosion . In thisway he found the following results of the comparative explosive force .Black gunpowderPicrate of potassaGuncottonNitroglycerineand for a mixture of explosives he uses the followingrule“Add the forces of each compound multiplied by thefraction representing its proportion in the mixture .If, however, this rule be applied to a mixture of ordinarygunpowder and n itroglycerine, the result will be foundmuchless than the actual result from experiment. From this itappears that the pressure of the more rapid explosionof an existing gun.4. Determination of the interior dimensions, of the conditions of loading, and of the powder to be adopted, to obtain134 INTERNAL RALLIsTICS.a given initial velocity, and maximum pressure, with givencalibre and we ight of projectile.5 . Determination of the interior dimensions and conditionsof loading in order to obtain, with the same powder, a giveninitial veloc ity and maximum pressure, in guns of differentcalibre.281 . These are the principal problems of internal ballistics,and the following is a résumé of the notation, employed.0 calibre of gun.W weight of projectile.length of travel.3 volume of powder chamber.w we ight of charge .A gravimetric density densité de chargementa, B, Characteristics of the powder.V initial veloc ity .P maximum pressure on base of projectile .P0 at breech.Unities, dec imetre, kilogramme, second.282. In the following calculationsf is taken 1 .When the problem involves the determination of thepowder to be used M. Sarrau adOpts the cubical form ofgrain, in which casea (if)?where a 31.where A 1 ,and the value of r , the duration of the combustion of a grainin free air, is the unknown which is to be determined.283. When the form is not cubical,the formula gives avalue -r'different from T , but it is generally useless to calculate 7'directly, when a previous calculation has given 7 .https://www.forgottenbooks.com/join136 INTERNAL BALLISTICS.286 , Admitting this relation the thickness of the grainis given by the formula(1-875 s)7287. When the powder used does not difi'er much fromthe types actually in service , the following method maybe adopted. Having found the value of r for a cubicalgrain, find the corresponding value of the characteristic1a (g)’and compare this with the values ofa of the usualTpowders as given in col . 4 of the table of Characteristics fordifi'erent powders given in If the value of a obtainedis contained between any two values of this table , thepowder to be adopted will be intermediate between the twopowders to which their value belongs, and this will generallybe suffic ient to determine the powder to be used.PROBLEM I.G ivena the calibre of the gun,I the length of travel of the projectile ,W the we ight of projectile,w the weight of charge,A the gravimetric density,To findV the initial veloc ity,P the pressure on the base of projectile,Po the pressure on the breech.Let a and Bbe the Characteristics of the powder.INTERNAL BALLISTIGS. 137289 . Then we havettV = AWe) (1 7)Yvllog A 3 -16767log B 2 1 8373.290. The value of cy must first be calculated. If it beless than the above formula is applicable , but ifgreater we must use the monomial formulaa w% A‘ cl l fl‘MV = aB'BWYZElog MSP1 powder in 90mm. gun.c 0-91 , I : IG° 7,A2 0 800, w s,w 0’ 857.1 95904 3 916383 70145502 ' 9 metres per second.138 INTERNAL BALLI8 TIUS.C l powder in 95 mm. gun.c = 0° 96 , l = 19 ° 6,s = 2 ° 640, W = 10° 9,w = 2 ° 1 , A = O° 795.3 ° 6l 9761 98227 4 10230log 7 3 ° 63749 458257° 43400monomial formula log V 3 ° 64405must be used. V 4406 dm.440° 6 metres per second.Calculation of Maximum Pressure.P pressure on base of projectile .P, pressure on breech .The formulae for the pressure are the followingA (Wm)iP = K a202on base of projectileon breechlog Khttps://www.forgottenbooks.com/join140 INTERNAL BALLI8 TI08 .Then from the above we findlogX1 or 1°or yThis being greater than the value of 1°may be admitted.2ndEaamp le.295. To determine a and B for the same powder fired in90mm. gun. Here the conditions arec = 0° 91, W = 8 ,w AV metres 5022 dm.From which we find1, g xor 7 , 0-6881 -38247 7 0-24125which being less than 02 73, the method of 254) must beadopted.First the value of V0must be obtained by formulamaking use of the value of To and 7 just found.This, in the present case , gives Vo 5058, then byformula2“ (I 7)(v v,)Vo (1 3 7)we findTo7which is the same as found in the first example .INTERNAL BALLI8 TI08 . 141PROBLEM III.296. Analysis of an eaisting Gun.Under usual conditions the chamber is not entirely filledwith the charge .The charge may therefore be increased, and combiningthis increase with a slower powder the velocity may beincreased without increasing the pressure .The amount Of this increase is however limited by thevalue Of the modulus, which, for reasons already givenshould not exceed297. TO appreciate the ball istic efi'ect possible to berealised in an existing gun with a fixed maximum pressure ,and with the above limit Of modulus,the following methodis to be adopted.Let P, be the pressure on the breech which must not beexceeded, and which is only bounded by the strength Of thegun. The relation (39) givesSubstituting 3:for A gives w as a function of P, and themodulus a.Giving a successive values decreasing by and beginning from the superior limit Of a 1 2 , we find a seriesOf weights Of charge realising the same maximum pressurewith powders Of increasing slowness.Then by formula (37)l f a I wI A’i ci llV A 8 B3 A wtwe get corresponding velocities. And since by (34)ta: 3BA (W l)1'6-r the corresponding time of combustion is found.142 INTERNAL BALLIsTIos .298 . The least value to be given to the modulus ise ither15, or that value superior to 19,which corresponds tothe maximum Of grav imetric density, which may be takenas unity.299. The formulas to be used areAl c e ll W P, -lf aallAvg'AI c ‘ liv A.W,1wherethe values off (a) and5being given in the table 323.24 cm. Naval gun Of 1870.s 35, W 144,2500kilog. per a 3,A 1 .301 . The following table gives the values Of-w, V, and 7 ,calculated as above,for values of a decreasing by 0 1downwards from 1 2https://www.forgottenbooks.com/join144 INTERNAL BALLISTICS.PROBLEM IV.Given the calibre and weight of p roj ectile, the in itialvelocity , andmaximumpressure, tofind the interiordimensions of the gun , the conditions of loading andthe powder to be used.The formula to be used aree Z EN—f)"“PM2K 2 (3 (it)7 K3A(Wcl)%‘iwhereThe values Of875 045W e“146 INTERNAL BALLISTICS.and the unity being the dec imetre, this corresponds to athickness Of 66 4 mm.Consequently, a powder Of similar manufacture toASfi, with a thickness Of grain of about 66 mm., will be thepowder required.PROBLEM V.Given the Weight of Projecti le, the Characteristics of thePowder ,the Initia l Veloci ty, and Maximum Pressure, to find theInterior Dimensions and Conditions of Loading in Gunsof Different Calibres.313. Let 0 and c’be the calibre .W and W'the weight Of projectile .V and P, the veloc ity andmaximum pressure .In the first place we must assume for the first gun calibrec,the gravimetric density A, and the modulus x, and find bymeans of formula the values of 3 ,f, and rW ewhich give for these guns the solution Of the problem. Wethen choose for the calibre c'a value Of the modulus x', andfind the corresponding values ofzWA' by the formula(86)314. The value chosen for the modulus should increasewith the calibre . Let c’c, then since the we ight of projectile is sensibly proportional to the cube Of the calibre , theformula and (89) are generally applicable, and ifwe suppose a x’these formula becomel'w'e W '1 (H)c Wand these give for the calibre c’a value of A', notably lessthan A and a value ofW" notably greater thanWwhichwould lead in general to an excessive size Of chamber.https://www.forgottenbooks.com/joinINTERNAL BALLI8 TI08 .c' 8Si nce5 7nea1 ly we may take x 0 7 andx'08 andfixing A at ° 950for the 27 cm. gun we find bythe application Of the above formulaFor the 27 cm. gun.wA Ww = 69 ° 6 , s = 73° 30.For the 32 cm. gun .A'w 126 6,s'172 -4.Supposing the grain to be cubical rIf the powder be Of similar manufacture to Asg-g, and thedensity 1 ° 820, the thickness of the grain as given by formula286) is about 42 mm.INTERNAL BALLISTICS. 149TABLE I.CHARACTERISTICS or FRENCH Powuuss.log B log a B' g log asTABLE II .DEsomP'n ON or THE n ova PowDERs.Density .Composition.Gravimetrlc . Absolute.6 2 to 6 8 8 toSaltpetre, 75Charcoal, 15Saltpetre, 75 518 to 14 l ° 800Charcoal,330 to t-385 to 1 133 1 -794104 t0 1 16 1-05 to 1 -15 l °78755 to 60 1 -1418 1 -15 o-911 8091 ° 800> l“738l °760> l°785> l° 8001 ° 815150 INTERNAL BALLISTICS .TABLE III.Dnm sxoxs or FBEROB Guns.Travel Weight ofProjec tile.Chamber.10cm. for the Marine ( 1870)80mm. L’dndGun,FieldSince 1 kilog. Of powder at gravimetric density densitéde chargement equal one occupies 1 dm. cube, the figuresin column 4 represent the weight Of charge when the chamber18 filled at A 1 .https://www.forgottenbooks.com/joinINTERNAL BALLISTI08 .TABLE V.— FUNCT IONS or THE MoDULus.los f (w) 108 v (as)REDUCTION TO ENGLISH WEIGHTS AND MEASURES.Value of Constants.324. In M . Sarrau’s investigations the unities are thekilogramme , dec imetre, and second. In order to make use ofhis formula with the English unities Of feet, lbs., and secondsit is therefore necessary to change the value of the constantsA,B, K, M, &c .The following table has therefore been prepared whichgives at one view the values of these constants in bothnotations.TABLE VI .Formula where used. sm og . 151mm .(43) (5 21 1)( 17) 177)( 13) (5 172)(60) 262)(64) (5 263)(90) 299)log A:P and P, . w andw.{5332 JET}English unities Tons per sq . in . PoundsINTERNAL BALLISTICS. 153Characteristics of English Powder .325. For ordinary English powder, it being Of the samecomposition as the C and SP French powder, the value Offmay be considered the same, and a and 71. being only dependent On the form of grain, the values of and therefore ofa. and B,may be determined for each powder in the mannerdescribed above .326. As regards prismatic powder, it is probable that somemodification will be required. The time of burning dependsupon the least thickness of the grain, which in a prismaticpowder is the difference between the radius across the flatsand the radius of the central hole . As these grains fit c losetogether in the cartridges, the first ignition is almost confinedto the surface Of the central holes, but as soon as the projectile moves, the grains separate , and then the whole surfacebecomes ignited. It is probable that this takes place beforeany considerable proportion of the charge is burnt,and if sothe duration of 7 will only be very slightly affected,in otherwords, the actual value of 7 will be slightly greater thangiven by the previous methods Of calculation.327. The value Off for prismatic brown powder and cocoapowder will probably differ from that for the black powders .The value Off is given (5 106) by the relationPo”0To273If then the values of r , and v, as given in 87 and (579)be admitted we should have for pebble powderx x 2230273and for cocoa powderx 198 x 2390154 INTERNAL BALLISTICS.Consequently, the value Off, as comparedwith unity adoptedby M. Sarrau, will be for cocoa powder7635.328. If this be so, itwould appear, that all other conditionsbeing the same, the velocity with cocoa powder will be87 3 2 per cent. of the veloc ity, with a like charge of blackpowder Of the same size and formOf grain.In order, therefore, to obtain the same veloc ity, it will benecessary to increase the charge, and as the veloc ity is proportional to the %th power of the charge , the charge of cocoapowder would be to that Of black powder, as 1 4 37 to 1, oran increase Of 43 °7 per cent.329 . These remarks must be taken with great reserve, asthe actual facts with regard to the temperature Of combustionOf cocoa powder are very imperfectly known .330. The characteristics a and B have been carefullydetermined for the French powders, so that,by means ofM. Sarrau’s formula, the ballistic results may be predictedfor any gun Of which the dimensions, the conditions Offiring,and the powder are known.331 . In one ofM. Sarrau’s works is given a table showingthe results of actual firing comparedwith those Of calculationsmade by the Binomial formula for slow powders, from e levendifferent guns, varying from 12 2 to inches calibre, witheleven different powders, varying from 13 grains to 800grains to the lb., or fromabout 1 inch cube to 13 inch cubes,and with gravimetric densities varying from 0 627 toOut of 40 rounds where the initial velocities varied from860 to 1960feet per second, 20 rounds averaged by calculation, 11 feet per second below, 14 rounds 12 feet per secondabove, and 6 rounds exactly agreed with the Observedveloc ities.332 . Again, calculating by the Monomial formula for quickpowders, he gives a table of 81 rounds firedwith ten differenthttps://www.forgottenbooks.com/join156 INTERNAL BALLISTI08 .INTERNAL BALLISTI08 . 157338. It is especially with regard to brown prismatic powderthat the above figures must be taken with reserve, as thedata in my possession are very scanty, and there is an additional source Of uncertainty regarding the value of f, as thecomposition Of these powders differs from that of the blackpowders.Similitude of Guns.339. Guns are termed similar when their lineal dimensionsare in the same proportion, and they are said to be similarlyloaded when the weights of the charges and projectiles are asthe cubes Of the calibre, and when the grain Of powder hasthe same form, is of the same composition, and has its leastlineal diameter proportional to the calibre .340. Under these circumstances, the initial veloc ities andmaximum pressure will be the same in all such guns.341 . The truth Of this proposition follows from an examination Of the formula (13) and (15) and it would be rigorouslyexact, except for one cause . The loss of heat fromthe absorption by the walls of the gun is proportional to the squareofthe calibre , and not to the cube , and as itmay be consideredas a reduction in the value of the charge, it will c learly berelatively greater in small charges than in large ones. Thetotal amount is, however, not very great, and therefore thegeneral principle Of similitude may be considered as true,with the reservation that it is not desirable to apply it toguns difi'ering very largely in calibre .342 . It is worthy Of remark, that in the construction Ofguns, and especially ofWire guns, as is shown by the formulagiven in my Treatise on the Application of Wire to theConstruction of Ordnance,’ the same princ iple Of similitudeexists, so that the dimensions and laying-ou tensions will besimilar in a gun Of l 2- inch calibre , to those Of a gun Of9 inch or 6 inch .343. In speaking of the laying-ou tensions being similar, it158 INTERNAL BALLISTICS.must be understood that the tensions Of laying on are thesame, at similar radii. For instance, if we have two guns ofcalibre c and c, and if c, mc, and if the tension at p be t inthe first gun, then the same tension will be applicable at aradius mp in the second, and the strains in the two guns underfire and at rest will be the same when the guns are similarlyconstructed and similarly loaded.https://www.forgottenbooks.com/join160 INTERNAL BALLISTICS.When gunmakers say, as they frequently do, that their gunswill produce a certain efl'ect,provided that a suitable powderbe found for it,’ they mean provided that the strength Ofthe powder be restrained, cribbed, cabined, and confined,to suit the weakness Of the gun .’ We sometimes see inhuman life a great and strong spirit tear to pieces a feebleframe which contains it, andwe do not say What a pity thatthe spirit is so strong,’ but rather, How sad that the body isso weak .’In the case of artillery we are always subduing andtaming the spirit instead of strengthening the body . Thismay be necessary under existing c ircumstances, but if so,the c ircumstances are unfortunate and stand in the way ofgetting the most value out Of the Spirit ofArtill ery .’A great deal has been said of recent years about thegreat improvement in powder, and it is held that thisconsists chiefly in its slow burning, and that still furtherimprovements may be looked for in this direction . IndeedGeneral Maitland, a few years ago , was so enamoured withthis view Of the subject thathe said,’ We find in the strugglefor existence,the guns growing longer and longer to get thebest effects from the slow powder, while the powder tends togrow slower and slower to meet the wants Of the guns, inaccordance with the eternal principle Of evolution ; and soimpressed was be with this view, that he said further, Alowmaximum pressure long sustained is the great desideratum of the artillerist,and no one will attain any measure ofballistic success who fails to recognise this fundamentalmaxim.”Again, Captain Noble, of Elswick, in a lecture at the Institution Of Civil Engineers in April 1884, said, When I addthat with a given weight of gun a higher effect can beObtained, if the maximum pressure he kept within moderatelimits, I trust I have said enough to vindicate the correctnessLecture at the R. U. S. Inst. on the Heavy Guns Of 1884, 2oth June ,1884, by Colonel Maitland, R.A.INTERNAL BALLISTICS. 161of the course which the gunmakers Of the world have, so faras I know, without exception followed.347. Now if by moderate limits Captain Noble meansa maximum pressure of about 17 tons per square inch,I would observe that the moderation must have reference to the strength of the gun, and I have no hesitationin saying, that, by the use of steel wire, a gun may be madewith the same margin of safety, under a pressure Of 30tons,as a forged steel gun of the same weight under a pressure of17 tons per square inch, and that the same ballistic effectcan be Obtained from the wire gun with amuch less chargeOf powder.Resistance to Bursting Strain.348. M. Sarrau’s formula enables us to determine themaximum pressure P, in a gun Of given calibre, with agiven charge and a required initial veloc ity.By the Binomial formula, applicable to slow,i . e . largegrained powders, it appears that as regards the first term, theveloc ity increases directly as the square root Of r , whilst inthe second term, which is subtractive, it varies inversely asthe 3power of r , so that the effect of increasing 7 is todecrease both terms, but the second, which is subtractive,more rapidly than the first, whilst by the Monomial formulathe veloc ity increases inverse ly as the one-eighth power ofConsequently with the same weight of charge the velocitymust always be less for a greater value Of that is to sayfor a large-grained powder, and the veloc ity must be madeup by an increased weight of charge, or by a greater lengthof gun.349 . As regards the maximum pressure, it increases asa“,i. e . as r , consequently the increase of pressure isrelatively greater than that of velocity.But the maximum pressure is only limited by the safeI162 INTERNAL BALLISTI08 .resistance Of the gun, and therefore it is clear that thehighest ballistic efl'ect must be obtainable from the strongestgun, whilst at the same time the gun (if a wire gun) neednot be strained relatively more than the weaker forged steelnu.g350.M. Sarrau has shown that there is a difference, whichmay be very considerable, between the maximum pressureagainst the breech and that against the base Of the projectile, and according to his formulaK0 AW5 witc2P = Kwhere P, and P are the pressures in tons as per square inchon the breech and base Of projectile respectively ;A,the gravimetric density ;0, the calibre in inches ;w andW, the weight Of charge and projectile respectively ;K and K, constants ;a , a factor depending upon the nature and formOf grain of the powder.Consequently the maximum strain on the powderchamber, determined by the formulaKo aQAWI tfic’(where K, is the strain to be provided for at thebreech end Of the chamber.351 . The strain P is not the strain against the base of theprojectile in its original position, but the strain when theprojectile hasmoved a certain distance at which the maximumis attained.This distance is not accurately determinable, but it maybe approximately found as will hereafter be shown. It ishttps://www.forgottenbooks.com/join164 INTERNAL BALLIsTIos.P, be fired in a gun with due attention to simultaneousignition, and the products then expand doing work upon theprojectile, the pressure will rise very rapidly, and will attainits maximum before the projectile has moved any considerable distance. With such powders it is probable thatthe whole Of the charge is burnt at the time Of maximumpressure, and that the work done on the projectile previousto that, is but a small proportion Of the total work done .Subsequent to the time Of maximum pressure, the curvewill be that Of a gaseous fluid, expanding and doing work,subject,however, to modification by the abstraction of heatby the cooling influence of the walls of the gun . Leavingout this modification for the present, the equation to thepressure curve will be given by Noble and Abel’s formulav, (1 a);zp + fl kv+ sa ’p pav a ”owhere p is the pressure corresponding to volume v ;the maximumpressure in a close vessel of volumevo fired at gravimetric density 1 ;Op , the specific heat at constant pressure ;C, , the spec ific heat at constant volume ;a , the ratio of the volume of non-gaseous products tothe volume Of the powder or v ;B, the ratio between the weights of the non-gaseousand gaseous products of combustion ;A, the spec ific heat Of the non-gaseous products.The values of these constants given by Noble andAbel arep 43 tons“per square inch 6554 atmospheres ;5 7 ;B 1 2957 ;Opo 2324 ;Ch° 1762 ;A.'45 ;This is probably greater when the weight of the charge is greater inproportion to the surface of the vessel in which the charge is burnt. Itmayalso probably be in some degree dependent on the composition of the powder.INTERNAL BALLISTI08 . 165and the above equation becomesPM“0A curve constructed from this formula, I call Noble andAbel’s curve, and if a point be taken on it corresponding tothe maximum pressure, the ordinates of this curve beyondthis point, will represent the pressure on the base of theprojectile at the moment when the projectile passes theseordinates.As regards the pressure curve previous to the time of themaximum, its equation is unknown, and as has already beenpointed out when treating of ignition,its determinationwould be of no great prac tical use .355. General Mayewski studied the question from experimental data obtained at Krupp’s works in 1867. The timescorresponding to the successive passage of the projectilethrough certain points in the chase were measured. Fromthese a formula was obtained expressing the Space as afunction of the time . From this by twice differentiatingthe acceleration andmoving force were determined.356. General Mayewski assumed a formula of the formand determined the coeffic ients so as to agree with themeanresults of experiment.Then by differentiation, he gotand by a second differentiation¢ = accel . forceis;The value of at corresponding to the maximum pressurewas given by the relationd’a:71730 6 0 + 24D t = 0166 INZ ERNAL BALLISTICS.Thus he found t '0018 and a: 4} inches, or theposition of maximum pressure was after the projectile hadmoved 4} inches.These experiments, however, were only made with a4-pounder gun, and with velocities of about 780 feet persecond.357. Captain Noble, of Elswick, made use of a somewhatdifferent method.He assumed a function of the form ofm a ta + fl t+ 7 t’and from the observed values of asand t he determined, bythemethod of least squares, the probable value of a , B,and ry,taking for unities n in th of a second andfi t} ; of a foot.The formula arrived at wasPebble{at 3 °31076 t 1°378 f’ Engl ish unities.POWder a: 1 0091 t 1'378 French unities .B L Ga: ° 57837 English unities.{as 1763 1534 2302 “ 9333“ French unities.358. Then by differentiation, the veloc ity and accelerating2force are determined. If this latter be 4: it“2celeration for mic-5th of a second, therefore the accelerationis 1000 and if W be the weight of projectile and P thePV:1000From this formula curves have been constructed showingthe veloc ity and pressure during the early part of themotionof a projectile of 300 lbs. fired from a 10- inch gun, withcharges of 70 lbs. P and 60 lbs. E.L.G. respectively, and itis said that these curves represent very approximately theactual results obtained.They show that whilst the maximum pressure is acquiredwith E.L.G . in 0 001 of a second, andwhen the projectile hasmoved about 05 of a foot, in the case of the P. powder, thetime was about0 0044, and the distance moved0 45 of a foot.this is the actotal pressure over the base,https://www.forgottenbooks.com/join168 INTERNAL BALLISTIUS.he in the case of the chamber being filled at gravimetricdensity I .Then, if the projectile be immovable and the charge firedthe pressure will be,according to Noble and Abel, about43 tons per square inch. Make AF the ordinate at A equalto 43,and then dividing AB into expansions of which A Cis the unit, set off along it ordinates such as D E,Y Y B0,representing the corresponding pressures as given by Nobleand Abel’s formula. The curve F EY1 C iswhat I callNobleand Abel’s curve .By Sarrau’s formula calculate Po and P, and set off 00,P0, and take the point E on the curve corresponding to Pand join 01 E. Then CEYI G is the curve of maximumpressure, and its ordinates give the values according to whichthe strength of the gun must be calculated, so far as regardsbursting strain .361 . I do not assert that the actual pressures in the gun arethose shown by the curve, but that the curve gives the superior limit, and is consequently a safe guide . The actualpressures in the chase will always be less than those shown bythe curve, as there is always a loss of pressure due to thecooling influence of the walls of the gun .362. There is,I think,cons iderable misapprehension onthis point, to which it is necessary to allude.Owing, probably, to acc idents which have happened to longguns firing large charges of prismatic powder, it has beenassumed that the pressures in the forward part of the chaseare much higher with slow than with quick- burning powder,and this is said to be due to the continued burning of thepowder all,or the greater part, of the time the projectile isin the gun.The re is no doubt that at times this does take place,and that in some cases a considerable portion of the chargeis blown out of the gun unburnt, but this is bad ballisticpractice, and I have already shown when treating of theignition andcombustion of powder, that in every case of slowcontinuous burning, the pressure at any point before the wholeINTERNAL RALLISTI08 . 169charge is consumed, must be less than the pressure at thecorresponding point in Noble and Abel’s curve .Nodoubt the pressures in the forward part of the chase aremuch greater now than they were in the days of quick powdersand shorter guns, but this is entirely due to the enormouslyincreased charges, whereby even a very long gun becomesvirtually a short gun , that is to say, a gun of few eXpansions,and it has nothing to do with the rate of burning of thepowder, which only afl'ects the maximum pressure in thevic inity of the chamber.363. This matter is so important,that, at the risk ofrepetition it may be well once more to explain it.Let A0 represent the charge of powder fired at gravimetric density 1 , andAB the length of the chase, andFEFIGNoble andAbel’s curve, as described. With a quickpowder, the whole of the powder would be burnt when theprojectile arrived at D , and the maximum pressure wouldberepresented by D E, and the pressures corresponding to themotion of the projectile fromD to B wouldbe representedbythe ordinates of the curve EG.With a very slow powder, the whole of the charge mightnot be burnt till the projectile had arrived at F when thepressure would be FF, exactly the same as the pressure from170 INTERNAL BALLISTI08 .the quick powder at the same point. Consequently thepressure on the chase between F andBwould be the same inboth cases. It does not follow that FF , would be themaximum pressure with the slow powder. Thatmight be ata point K where the increasing evolution of gas was exactlybalanced by the increasing space behind the projectile, but atthis point, and at every point between K and F,the pressuresmust necessarily be less than with the quick powder for thesimple reason that there is a less quantity of powder gas inthe same space .364. It follows, therefore , and this is the important point asregards gun construction,thatwith equal charges the pressuresfrom slow powders must generally be less, and can never begreater than from quick powders,and that consequentlyNoble andAbel’s curve is a safe guide to the gun constructoras far as the bursting strain is concerned.365. Since the ordi nates of the pressure curves as derivedfrom Noble and Abel’s curve, represent the pressure persquare inch on the base of the projectile, the area of suchcurves multiplied by the area of the base of the projectilewill give the energy of the shot, and if there were no loss bycooling and nothing expended in the friction and expulsionof the gases, &c ., the muzzle veloc ity would be obtainablefrom the formula366. The actual pressure curve as regards the projectilemay be obtained fromSarrau’sformula for initial velocity ;for if v be the initial veloc ity ;4) the accelerating force ,W the weight of the projectile in lbs. ,or the area of its base,I the travel of the projectile,V dVd l4”https://www.forgottenbooks.com/joinINTERNAL RALLISTI08 .° 21316 log ,8 1 -93843c calibre 6 inchesW 100fromwhich we finda 1 159 and b 1023V 1 159 li 1 18 llandmaking I the total travel of projectile we findV 1875 feet per second, which was very nearly theobserved velocity.368 . For the pressure per square inch at any intermediatepart aswe have, making I as24 785wtFromwhich the curve in the following diagram (Fig. 1)has been obtained.XThe pressures calculated from Sarrau’s formulas (17) and(15) are8 1 :43:13iP0 tons per sq . inch .Pthe latter corresponding to a travel of Projectile of 1 } inches.The actual pressure given by the crusher gauge was about25 tons per square inch.The area of the curve is 83 3, which multiplied by 28the area of the projectile gives the energy 2332 foot-tons,therefore1834 feet per second,which agrees very nearly with the cal culated and theobserved velocities.369. The upper dotted line in the diagram shows thepressure curve according to Noble andAbel’s formula,and itwill be seen that it is throughout higher than the curve fromSarrau’s formula. Taking the area of the upper curve asrepresenting the total energy, that of the lower one theINTERNAL RALLISTICS. 173energy expended on the projectile in giving velocity, it will befound that the latter is about 76; per cent. of the former,showing that about 23; per cent. is expended in expelling thegases, friction, &c ., which, as will be seen in the last chapterof this book,is probably very near the truth.370. As a second example I will take a lO- inch gun witha projectile of 500 lbs. and charge of 300lbs. prismatic brownpowder.174 INTERNAL BALLISTICS.In this casesv 752 13 - 19 17 3,and when lV 2125 feet per second.The actual velocity observed was 2100feet per second. Theequation for the pressure is found to bep 1 -6 fl 0 289E,fromwhich the following diagram (Fig. 2) is obtained.Themaximumpressures calculated by Sarrau’s formula areP0 17 80P 10 50.The pressure by the crusher gauge was 18 tons. The energycalculated from the curve is foot-tons, which corresponds to a veloc ity of 2083 feet per second, which was verynearly the observed velocity.371 . The upper dotted line shows the curve fromNobleand Abel’s formula for pressure . It will be obse rved thattowards the muzzle it falls slightly below the other curve .This is no evidence against the truth of Noble andAbel’scurve . It arises from the value assumed for the index 9”in192) being a mean value approximating from the ordinarycondi tions offire ; and when the weight of charge was about5rd of the weight of projectile. The formula is thereforeonly approximate under the conditions of the second examplewhen the charge was gths of the weight of projectile. Thisleads to a change of the form of curve, although it does notappear to cause any important error in its total area. Calculating the energy from the curve it is found to befoot-tons, so that in this case the percentage of energy spenton the projectile is 81 15 per cent.,and on the expulsionof gases, friction, and cooling, about per cent.and we findhttps://www.forgottenbooks.com/joinINTERNAL RALLISTI08 .observed muzzle velocity to the projectile, as well as toovercome the other internal resistances and the coolingaction of the walls of the gun, and consequently a gunwhose strength at each portion of the chase corresponds tothese pressures, and to the maximum pressure in thechamber as determined by Sarrau’s formula, will always bea safe gun as regards the bursting strain.Longitudinal Strain.373. There is, however, another strain to be provided for,the Longitudinal Strain.So far as I know,the only longitudinal strain which hasbeen considered important by artillerists and gun-makers isthe strain between the breech and the trunnions.The maximum value of this strain is at the obturatord if the gun be supposed to be fixed at the trunnions so asto have no recoil, the amount of this strain is Pomwhere P(,is the maximum pressure and a) the sectional area of thechamber, and in the case supposed of no recoil, the samestrain extends to the trunnions.If, however, the gun is free to recoil, the strain will begradually diminishedby the force requiredto give accelerationto the mass behind the point at which the strain is calculated,and in this case the strain will be one gradually dec reasingfrom the obturator to the trunnions.374. The usual way of dealing with this strain is to assumethat it is uniformly distributed over the cross-sectional areaof the gun. This assumption is entirely wrong. The strainat the obturator is borne unequally according to some lawwhich is not accurately known, but there is good reason tobelieve that it is analogous to the law which governs thebursting strain, and that it varies inversely as the square of thedistance from the axis of the gun. This being so, the innersurface is strained very much more than the average strainon the whole cross-section.INTERNAL BALLISTI08 . 177For many years this was disregarded by gun-makers, andthe evil was aggravated by throwing this strain directly onthe inner tube of the gun, and from that to the breech coilor jacket, through which again it was carried to the trunnions. The condition was still further aggravated by thefact that this portion of the material of the gun had also tosustain the bursting strain. There were consequently twoconjugate strains, each of great intensity at the inner surfaceeach to be borne by the same material.375. So long ago as 1860 I drew attention to this, andadvocated the entire separation of these strains, but noregard was paid to the matter, and it is only within the lastfour or five years that my views have been partially adoptedin breech-loading guns by making the breech screw takeinto the jacket instead of as before into the tube . This is,however, a very partial improvement, inasmuch as owingto the thinness of the inner tube at the breech end, thejacket has to resist a very heavy bursting strain .376 . My op inion has always been that the whole of thebursting strain should be exclusively borne by the tube andits reinforcement, and the whole of the longitudinal strainexc lusively by the jacket ; and in my very first paper in 1860I showedhow thismight be done in the case of wire guns, andyet till within the last few years it has been persistentlyasserted that this was the greatest difficulty as regards theconstruc tion of wire guns. The fact that I had shown howthe longitudinal strain was to be provided for, that I hadactually done it in a gun of which the inner tube was of castiron and only half an inch thick, was quietly ignored, and itis still constantly asserted that the great difficulty in wiregun construction is to provide for the longitudinal strain.It has been said by some that my system involves extraweight, that the jacket forms a considerable part of theweight of the gun , and that therefore its material ought tobe utilised in inc reasing the resistance to bursting strain,just as if the material would resist the action of two conjugatestrains of equal amount to its tensile strength !178 INTERNAL BALLI8 TI08 .When the strains are kept separate, the prec ise amount ofeach being known, the provision by distinct members isaccurately dete rminable, but when they are mixed up so asto act conjointly on one and the same mass ofmaterial, as atthe breech end of a gun, the problem is exceedingly diflicult,d perhaps practically insoluble.The princ iple of the separation of the two strains istherefore , in my opinion, one of primary importance.LongitudinalStrain before the Trunnions.377. The longitudinal strain between the trunnions andthe muzzle does not appear to have been considered ofmuchimportance, and yet of late years many cases have happenedin which a portion of the chase in front of the trunnion hasbeen fractured.378 . For instance, the 12-inch Coll ingwood gun, whichon l0th May, 1886 , blew away about 8 feet of the front ormuzzle end of the chase.A Committee of Investigation was appointed, consistingof the Ordnance Committee , with whom were assoc iated theSuperintendent of the Royal Gun Factory andmembers ofthe Elswick and'Whitworth Ordnance Factories. After a longinvestigation they concluded that the accident was due tol st. Want of uniformity in the metal.2nd. Absence of annealing after forging and hardening,and resulting internal strains.3rd. Intensification of such strains by firing at proof,andfurther self-development during the interval of eighteenmonths between firing at proof and the acc ident.4th . The want of chase-hooping.They recommended that in these and all other guns of6- inch calibre and upwards the chambers should be reduced,the service charges reduced, so that the maximum pressurein the chamber should not exceed 15 tons per square inch,and that the chase should be hooped all along to the muzzle .379. From these recommendations it would appear thathttps://www.forgottenbooks.com/join180 INTERNAL RALLISTI08 .be seen that the tube and coil were entirely free to movelongitudinally in the jacket.The jacket was of cast iron,and at the muzzle end a steelring was screwed on, which had a deep flange projectinginwards, against which the steel tube and coil abutted. Itwas this flange alone which prevented the tube movingforward,and consequently any longitudinal forward forceacting on the tube was borne by this flange , and thus transmitted to the jacket at the point A.At the breech end of the tube, and between it and thebreech ring screwed into the jacket, were six set pins,keptup against the flange of the tube by Belville springs, witha forward strain of three or four tons, so that whilst the tubewas always kept up to the flange at the muzzle it was freeto expand backwards against the springs.The breech block, it will be seen, is entirely independent ofthe tube and coil,so that the bursting strain is entirelyprovided for by the tube and coil, and the longitudinalstrain by the jacket.381 . In designing this gun I did not lose sight of thefact that a considerable longitudinal strain would be thrownon the jacket between the muzzle and the trunnions. In thefirst place there was the pressure of the powder gases on thedifl'erence of area between the chamber and the chase,about1; square inch . Then there was the friction of the projectileon the grooves, or rather its resultant in the direction of theaxis of the gun, and lastly, there was the inertia of thetube and wire coil which had to be set in motion backwardsas the gun recoiled. The whole of these strains I had calculated and amply provided for, and yet when the gun wasfired the jacket was torn asunder at the point A, and thesteel muzzle ring projected violently to the butts.382. The sectional area of the jacket atAwas 55 °8 squareinches, and the test pieces cut fromit gave a tensile strengthat rupture of 16 tons per square inch . If, however, only onehalf of this be taken, the rupturing force must have been446 tons. As will be shown hereafter, the utmost strainINTERNAL BALLI8 TI08 . 181that could arise from the sources abovementioned would notexceed 284 tons, leaving a force of at least 162 tons to beaccounted for.Where didthis force come from On examining the gunand projectile it was found that there had been no jamming,in fact the projectile had been passed through the gun byhand previous to firing, and when fired it went straight tothe butt, with a veloc ity of about 1870 feet per second, thefull veloc ity calculated for the charge of 34 lbs. P. powder.The tube was uninjured, but it had moved bodily forwardnearly an inch in the jacket. It was therefore evident thatit was the forward motion of the tube which ruptured thejacket, and that that forwardmotion relative to the jacket hadbrought into play a force of at least 446 tons, of which Icould only account for 284.383. After a little consideration I came to the conclusion,which subsequent examination of the subject only confirms,that a very important forward longitudinal strain,hithertoaltogether unapprec iated, was caused by the friction of theproducts of combustion against the inner surface of the tube .So strong was my conviction of this that a few days after theacc ident I wrote to the War Oflice proposing that the gunshould be sent back to the makers to be repaired, and that Ishould have an interview with the Ordnance Committee toexplain to themmy views and reasons, before they reportedon the acc ident. I also suggested that experiments, thenature of which I was prepared to explain, should be madeto set the question at rest, which was the more desirable,inasmuch as it appeared tome to have an important bearingon the future of the Collingwood and other guns.No notice was taken of my'letter. I was not allowedto see the Ordnance Committee, but after about twomonths, having heard privately that they had reportedon the acc ident, I wrote again to the War Ofii ce, askingwhat was going to be done with the gun, and that I mighthave a copy of, or be informed of the reasons given bythe Ordnance Committee for the accident in their report.182 INTERNAL BALLISTICS.The reply to this was that no further expenment would bemade with the gun, and that the report was a confidentialdocument, and I could not be allowed to know its contents !I have , however, learnt that the Ordnance Committeeentirely rejected my view regarding the friction of theproducts of combustion, and I have 110 reason to supposethat my suggestion of experiments being made is likely tobe entertained.384. Under these c ircumstances, andwith adeep convictionof the importance of the question, I now submit the followinginvestigations, not as a solution of the problem, but in thehOpe that itmay be the means of directing the attention ofothers to the subject, who may be much more competent todeal with it than myself, and who may have the means ofundertaking such experiments (which are not of a difficultnature), so as to ascertain prac tically what is the real effectand amount of the friction of the products of combustion onthe surface of the chase of a gun.385. In the first place I will make a few brief remarks,which will perhaps be admitted to contain primai facieevidence that the amount of this friction may be very considerable .386. The erosion of guns is the work of the products ofcombustion. These products are not altogether gaseous, alarge proportion is generally admitted to be liquid in a verydiffusedstate, but however thismay be, there is no doubt thatthe mixed products do actually cut away particles of solidsteel as though they themselves were solid. This cannot bedonewithout exerc ising a forward force upon the exposed surfaces, because these products are themse lves moving forward.387. Let the gun be considered at the moment of maximum pressure . At this moment the projectile will havemoved forwardabout 10inches. The surface exposed wouldbe 636 square inches, and the pressure 25 tons per squareinch. There is therefore amass of mixed gases and liquidpressing on a surface of steel, with a pressure of tons.At the breech end the mixedmass is at rest, but against thehttps://www.forgottenbooks.com/join184 INTERNAL RALLISTICS.forwards. Either case is quite possible, and, in fact, chaervations tend to show that in some cases the recoil is actuallyprevented until the projectile leaves the muzzle.390. In an interesting paper by Mr. H. J. Butter,of theRoyal Gun Factory, Carriage Department, read at the Institution of Civil Engineers, 22nd November, 1881, it is statedthat From results obtained by instantaneous photographyin connection with the firing of a 25-ton gun, using E.L.G .powder, it was shown that the shot was just clear of themuzzle before the gun moved. More recently it was asoertained by electric ity that, in the case of the 6-inch B.L. gunusing pebble powder, the shot was within 2 inches of quittingthe muzzle when the firstmovement of the gun occurred.”391 . Now, if there were no opposing force, there could beno doubt that the gun would begin to move very nearly atthe same instant as the shot, and setting aside the friction ofthe carriage in the slide, on the one hand, and the frictionof the projectile on the other, the veloc ity of the recoil would 'be to that of the shot, inversely as the respective massesmoved ; and if these be, for example, as 100 to l , and themuzzle velocity 1800 feet per second, the gun must haveacquired a velocity of 18 feet per second backwards whenthe projectile left the muzzle. But we are informed that inreality it had not moved at all .392. Again, it is stated by Major Mackinlay, RA , in hisText-book ofGunnery,’ 1887, that by aFrench experimentit was found that a 24-cm.- inch) gun had only recoiledi nch during the time that it had taken for the projectile to travel all the way down the bore ; the veloc ity ofrecoil was thus feet, and it attained its maximumveloc ity of 17 feet at a period second later.” In thiscase the gun was a comparatively short one, with amoderatecharge of powder, so that the effect of friction in producingthe immediate recoil was not so greatMr. Butter’s statementmust be taken with reserve. It is quite possiblethat there are cases in which the gun does not move backwards until theprojectile leaves the muzzle , but that is not always the case .INTERNAL BALLISTICS. 185393. Mr. WilliamAnderson observed that in the case ofa l O- inch gun fired from a Moncriefl'disappearing carriagethe gun did not move at all till the projectile left the muzzle.394. There is therefore,I think, very strong presumptiveevidence of the existence of important longitudinal forwardstrains between the trunnions and the muzzle , the magnitudeand perhaps the existence of which have not hitherto beensuspected.395. I now proceed to the investigation of these strainsas follows1 Strains due to Inertia of Mass infront of Trunnions.396. Let M total mass of gun and recoiling part ofcarriage ;m mass of the portion of the gun in front ofany section at a ;R oute r radius of chase at a ;p radius of bore ;P maximum internal powder pressure.Then the moving force is P vrp2 M 5332, and the movingforce acting at the section at a will beand taking the weights W andW1 , and observing that thesectional area at a '11'(R2 we getW; P P2Stramper sq. mch eta: W (R p2)°2. Forward Strain due to the Difi'erences between the Area ofthe Obturation and that of the Bore.397. This is equal to the maximum pressure multipliedby the difl'erence of area, and it may be taken up at. thetrunnions, in which case it only acts as easing the recoil .186 INTERNAL BALLISTICS.3. Forward Strains due to the Projectile .398. This strain is threefold. First, there is the strainrequired to force the rotating ring into the grooves. Itsamount depends, of course, upon the dimensions andmaterialof the rotating ring,but it is probably of no great amount,and only acts for a short time at the beginning of the motionof the projectile, retarding its motion and increasing themaximum pressure . Its effect may therefore be consideredas included in that of the pressure .The next part of the strain is that due to friction . Thefriction of the projectile on the chase considered as a slidingfriction is simply the we ight of the projectile multiplied bythe coeffic ient of friction, and is so small that it may beneglected. There is, however, the fric tion arising from thereaction of the rifle grooves against the projectile, whichmay be estimated by the well-known formula.Strain ar isingfromFriction of Products of Combustion.399. There are two hypotheses by which this subject maybe investigated.First, that the resistance due to friction varies directly asthe density and as the square of the veloc ity of theproduc ts.Second, that it is independent on the velocity and variesdirectly as the pressure .l stHypothesis.400. Let it be assumed thathttps://www.forgottenbooks.com/join188 INTERNAL BALLISTIGS.Now the density atand the surface is 2 vr p d y, consequently2 1rzgp dyvsg- 1 31 +w“)and writingR = f Bctr/+ 011 3!1 (w 1i )"J’(L at w, {5 + 13, Constant} (6)When y 0 the resistance is that due to the surface ofthe length I, but then the veloc ity is zero,thereforeR 0, and Constant 0, and when y aR = f B (xand when a LSecond Hypothesis.Adopting the same notation and making use ofNoble andAbel ’s formula, if p , be the pressure on the baseof the projectile at any point a4 3 1P ia: ° 57 llbut the pressure at the breech is always greater than that onINTERNAL BALLISTICS. 189the projectile, therefore let it be denoted by a p ,” then themean pressure(2)The area of surface exposed 2 71 p a, and if 4; be thecoeffic ient of friction,R= 2 1r pwp £p3 = P° 43 lcc 5° 7 lNow P is a maximum for some value of a a, greater than Iwhich will be found by the relation° 43 lP’= Pwl° 57 lWhere P1 is the maximum powder pressure in the gun, andP the pressure in a close vessel 43 tons,fromwhich we get(gym4 31+oa7 l ,introduc ing which value of a into (3) we get the maximumresistance-57 1} (5)App lication to 6-inch Gun.l stHypothesis.L feet.I 2 833C 0° 551 cubic feet.34 lb34w : s =224ofromwh1ch6656and by formula (7)R 4746 f.190 INTERNAL BALLISTICS.403. Now, if we take the rupturing force as 446 tons, asshown in 5382, we have(a) Strain arising from inertia of the"tube and coil, asfollowsVV’P n-p22 ° 27 tons.5 tons per sq . inch .8 sq . inches .(ONOTherefore,Strain x 25 x 28 244 tons.404. (b) Strains arising from difference of area ofobturatorand bore25 tons 38 tons .405 . (0) Forward strain due to projectile .In this case the projec tile had no rotating band, therotation being given by ribs, and the twist was a uniformtwist of 1 in 30.The coefficient of friction n is taken as §th, and the workdone to overcome friction would be1 737 p”P l"237 1237 u m (L‘ 43Ow land sincep P 25 tons per sq . inch , 1 :L m: 30, 73Work done 15 79 foot-tonsMean force 1 °504 tonsMaximum force 11 ° 66 tons ;https://www.forgottenbooks.com/join192 INTERNAL BALLISTIOS.coeflicient of friction as depending only on pressure,these cond being a coeflicient which varies with the density andveloc ity of the products.410. Further confirmation of the longitudinal strain dueto friction of the products is given by the fact that in a6 - inch wire gun, made from my designs at Aboukoff,byAdmiral Kolokoltzofl', the part of the steel tube in frontof the trunnions, about 8 feet in length, was actually elongated 135 of an inch after firing 500 rounds.This gun differed from the gun which was fired at Woolwich,inasmuch as the greatest part of the forward strain wastaken up by the jacket a little in front of the trunnions,where the sectional area of the jacket (also cast iron)was verymuch greater than that at the muzzle of the Woolwich gun .The e ight feet of the tube in advance of this was quite freeto elongate, and there was no force ac ting on it except thefriction of the projectile and that of the produc ts of combustion .It could not be elongated by any deformation arising fromthe internal pressure, as the pressure on that part was veryfar below the elastic limit for compression.411 . The foregoing observations and investigations mustbe taken with reserve , but I think they afford strong presumptive evidence that this hitherto neglected e ffec t of thefriction of the products has a very important bearing on thelongitudinal strain on the chase of a gun in front of thetrunnions .The acc ident to the 6 - inch gun referred to inwas certainly due to a forward force which I am unable toaccount for otherwise , and I think it not only unsatisfactory,but very unfair, that I was neither allowed to give my viewsto the Ordnance Committee , nor to see the report whichthey made to the War Office. To treat this report as confidentia as between myself and the Ordnance Departmentis not only absurd, but it is unbusiness- like and unfair, andit is di fficult to understand how such reserve can be beneficial to the public service.INTERNAL BALLISTIOS. 193Chambering.412. A good deal of importance has been attached tochambering, as if it were a means of reduc ing the maximumpressure in a gun, without affecting its initial veloc ity .This it cannot do. The maximum pressure, caeteris paribus,varies as the gravimetric density, which has nothing to do withthe diameter of the chamber. The only effect of increasingthe diameter of the chamber is to decrease its length for thesame charge of powder, and of course this decreases the totallength of the gun, but, as may easily be shown,does notdecrease its we ight. In fact it somewhat increases it, owingto the extra size of the breech plug. The muzzle veloc ity,cwteris paribus, depends on the length of travel of the projectile, so that the actual length of the gun is decreasedsimply by the difference between the length of the chamberand its equivalent length. This decrease of length is doubtless an advantage for naval guns .413. There is no difficulty in constructing chambered wireguns, and I am of opinion that a 12-inch chamberedwire gunof about 65 tons in weight and 30 feet long, might be madewith equal power of penetration as the 1 10-ton inchElswick gun at 1000 yards, and which would exceed it atany longer distance, working, of course, at a higher pressure,but with no greater relative strain on the gun.414. As regards the alleged wave pressure, I will onlyrepeat what I have shown before when treating of Ignition,that with a properly constructed cartridge, so as to insurerapid Ignition, there will be no wave pressure .415. The unchamberedgun has however the advantage thatany weight of charge can be fired with full gravimetricdensity, while in the chambered gun any charge below thefull charge . must be fired with lower gravimetric density,depending on its weight, and consequently with a loss ofballistic efl‘ect .194 INTERNAL BALLISTIOS.Rifl ing.416. The system of rifling has little or no effec t on themuzzle veloc ity of the projectile or the maximum powderpressure . As regards accuracy of fire , it matters nothingwhether the twist be uniform or increasing, provided theincreasing twist ceases to increase a short distance from themuzzle . If in the last three or four calibres of length theprojectile has the right amount of rotation and is trulycentred, which it may be with either system of rifling, itmatters nothing, so far as accuracy of fire is concerned, whatwent before, or how that rotation was acquired.The danger of over-riding the grooves, or of any partialjamming, is greatest with the uniform twist at the beginningof the motion and decreases very rapidly as the projectileacquires velocity , whilst with the increasing twist it is exactly the reverse, consequently the effect of any such partialjamming, depending as it does on the sudden loss of energy,is much greater with the increasing twist.417. I will conclude this chapter by a few words on thissubject. I have already given my reasons, when dealingwith the powder question,for thinking that the very seriousincrease of erosion is due chiefly to the enormous increase inthe volume of the products of combustion, consequent on theincreased charges required by low pressure powder and to thevery high temperature of combustion of the brown powder.The Russian 6-inch wire gun above mentioned fired 1000rounds with 394 lbs. of black prismatic powder, and theerosion wasmoderate, and the gun still serviceable,althoughof course the accuracy of fire was diminished. Generalhttps://www.forgottenbooks.com/join196 INTERNAL BALLISTICS.appealed to, to show the great superiority of the wire systemto that of h00ps. I wish, however, to call attention to theremarkable work of General Kalakoutski on the internalstrains inherent in cast- stee l forgings, and the dangerouscharacter they may assume under the ordinary practice ofhardening, tempering, and annealing, and the aggravationof the danger whichmay arise from the minute inaccuracies ofworkmanship wh ich it is impossible to avoid in such a complicated structure as a large hooped gun . From all thesedifli culties the wire system of construction is free, and I feelconfident that the views on this subject which for the lastthirty years I have been advocating, will ere long be admittedto be correct even by those who have so - long looked uponthemas the dreams of a visionary t heorist.INTERNAL BALLISTICS. 197CHAPTER V.GUNS CONSIDERED AS THERMODYNAM IC MACHINES.420. Count St. Robert in his Princ ipes de Thermodynamique,’Turin ,1870, indicated generally the relation ofballistic effect to thermodynamic laws, and in a paper presented to the Institution of Civil Engineers,and publishedin vol. lxxx., Session 1884—85, of their Minutes of Proceedings,’ I endeavoured to apply Count de St. Robert’smethod to determine the initial velocity and velocity ofrecoil of rifled guns.421 . The subject is one of considerable interest, and asthe paper above mentioned contained many typographicaland other errors, 1 have modified it in the present chapter.422 . A gun is a machine for the conversion of heat intomechanical force, just as much as is a steam- engine. In thegun the heat operates through the mechanical force of expansion of the gases, which are evolved simultaneously withthe heat.These gases pass through a cycle of which the initial stateis the high temperature of ignition of the powder, and thefinal state that when the projectile leaves the gun. Consequently the heat expendedmust be represented by the workdone in overcoming the various resistances, and in imparting energy to the projectile, the gun, and the products ofcombustion.423. LetAH be the units ofheat abstractedfrom the productsof combustion in passing from the initial tothe final temperature .198 INTERNAL BALLISTIOS.AQ, the units o f heat passing from the products intothe body of the gun.A l , the increment of internal work in the gasesduring the same time.AW,the external work done, consisting of(a) The statical resistance of the air tothe motion of the projectile , inother words,the atmospheric pressure, not inc luding the increasedresistance due to veloc ity.(b) The work done in giving rotation tothe projectile.(c) The friction of the projectile on therifling.(d) The work done in overcoming thefriction of the gas- check and projectile .(e) The work done in overcoming thefric tion of the produc ts of combustion in the bore .(f) Work done in stretching the gun c ircumferentially and longitudinally.AV the sum of the energy acquired by the wholesystem of projec tile , gun, gun - carriage, products of combustion, and s the resistance of the air due to the veloc ity of theprojectile.J, Joule’s coeffic ient of the mechanical equivalent ofheat,or772 foot- lbs. 1 unit.424. Then we have the followmg relation ,https://www.forgottenbooks.com/join200 INTERNAL BALLISTI08 .estimated that about 250unitsof heat were imparted from~one kilogramme of powder, or about 30per cent.Noble and Abel, from experiments made on a 12-pounder3- inch gun, found that 545 units of heat were abstracted in9 rounds of 14 lbs. each, or 7 ° 2 kilogrammes, which gives605 units per round, or 75} units per kilogramme of powder,01°about 9 6 per cent.They further estimated that with a 10- inch gun the losswould not exceed 35> per cent.427. There can be no doubt that the percentage of loss ismuch less in large guns, because whilst the absorbing surfaceonly increases as the square of the lineal dimensions in similarguns similarly loaded, the weight of charge, and consequentlythe quantity of heat evolved, increases as the cube .Without pretending to any great accuracy in a questionwhere the many elements, or some of them, are uncertain, anapproximate solution of the problemmay be attempted.428 . In it was shown that if s be the loss oftemperature,log z + z log a = logwhere a constant ;°000237emissive power of gases,assumed 1absorbing power ofmetal ° 15 ;mean spec ific heat of products of combustion186 ;mean temperature of products ;surface exposed in dm.2weight of charge in kilogrammes.W ith these values (4) becomeslog 2 log-0001910 x 1 -0077To x f } log 2 (5)fromwhich 2 is easily determined.Then 2 x 0 units of heat represented by the fall ofINTERNAL BALLISTI csftemperature, and if the total units evolved from1 kilogrammebe taken at 728,percentage of l oss.429 . The above expression (5) contains 7 and To. Theformer is the time during which the gun is exposed tothe heated products, whilst the projectile is passing along thebore . Th is may be estimated, if we know the length ofthe gun and the muzzle veloc ity, by means of M. Sarrau’sformula.In the case of a quick powder this formula is of the formV = ofi .If, therefore, we represent the veloc ities by the ordinatesof a curve , of which the corresponding absc issa; represent thedistance travelled by the projectile , the area of this curve ,taken from l 0 to l l (the length of travel) divided by l,will give theMean veloc ityand sincewe getMean veloc ity° 842l V .Therefore the time which the projectile takes in reachingthe muzzle isand sinceVzr’swe get finally the time in reaching the muzzlel ° 188 E.V202 INTERNAL BALLISTICS.430. The temperature of the gun varies as the projectilepasses along the bore , but it will be suflic ient for the presentpurpose to take To as the mean temperature during the timethat the projectile is in the gun. This may be found asfollows:431 . According to Noble andAbel’s formulat and to be ing absolute temperatures, or, as shown before,074Since the volumes are proportional to the distance passedover,taking for v, the equivalent length of the chamber,that is to say, the length of a cylinder whose diameter isthat of the bore and whose capac ity is that of the chamber,and making this equivalent length the unity of length, and ythe distance moved by the shot measured in the same unity ,the value of v will be v, y 1 y, and432 . If then the temperature be represented by a curve ,the area of that curve divided by the length of absc issa willgive the mean temperature . But the area of the curve is[L ~ 9394 t,where L is the length of the gun inc luding the equivalentlength of the chamber, and l is the equivalent length ofchamber, or if l be taken as unity, L is the number ofexpansions.https://www.forgottenbooks.com/join204 INTERNAL BALLISTICS.and it is, moreover, liable to uncertainty with respect tothe value of the emissive power of the produc ts of combustion. It is probable, however, that in the case of largeguns, with heavy charges of slow- burning powder, the approximation is tolerably correct.A I. Internal Work of Gases .436. The internal work in a perfect gas expanding iszero, and as powder gases approach very c losely to the condition of a perfect gas, we may assume A . I 0.AW. Es ternal Work.437. (a) The first item of external work is the work doneagainst the atmospheric pressure,apart from any increase ofresistance due to the veloc ity of expulsion .If therefore p atmospheric pressure ;A area of bore ;L length of travel of shot ;R resistance ;the work done RL p .A .L ;Work done in giving Rotation to the Shot.438 . (b) Let W weight of shotm number of calibres to one turn of theshot or twist 1 in m;V muzzle veloc ity of shot ;707 radius of gyration for cyl indricalbody revolving round its axis.Then the velocity of rotation of the centre of gyration is“707 W Vand the mass isEm 9therefore11. (7072 g mINTERNAL BALLISTI08 . 205Fr iction of the P roj ectile in the Groove.439. (c) For the sake of simplicity, the twist is assumedto be uniform. Then if the pitch of the rifling be 1 in mandP the pressure on the base of the shotForce to give rotation P 15)If p be the powder pressure at any point asof the travel ofthe shot, the pressure on the base will be p qr p”, p being theradius of the base, and substituting in we get2Force to give rotation at a:"ap2mIfPl be the initial pressurep P1 21 . 57and force to give rotation at a(w.P), e 432 m1and 1f5be the coeflic1ent of fr1ct1on, and the pressure he intons per square inch, this reduces to1 p’ 1 ° 737 P,n m(z, . 57)1 °2a7Now if l he the equivalent length of the chamber in feet,and a the area of the bore206 INTERNAL BALLISTI08 .Work done in overcoming the Grip qf the Rotating Ring when440. (d) The friction prOper of the shot, as distinct fromthe friction due to the reaction of the rifled grooves, is onlyWnwhen W is the we ight of the shot, so that the work doneYV—L which is so small that it may be neglected. Thenforce required to press the shot into the grooves is only atthe beginning of the motion, and is only a small fraction ofthe powder pressure , and as this ceases as soon as the ringhas entered the grooves, the work done is quite insignificant,and may also be neglected.Work done in overcoming the Friction of the Products ofCombustion in the Bore.441 . In a previous chapter the question of the efl'ect ofthe friction of the products was discussed on two hypotheses,with reference to the longitudinal strain produced on thechase of the gun.In now attempting to ascertain the work done in overcoming this friction I will adopt the second hypothesiswhich supposes that the friction is proportional tothe pressure.442. When the projectile is at any point a the resistancedue to the friction is by (3) 401)41 being the coeffic ient of friction.Now the front portion of this mass of products movesforward da, whilst the rear portion is at rest. Whereforehttps://www.forgottenbooks.com/joinINTERNAL BALLISTICS.For any other extension 2,less than a,lot 41 be the forceexerted, then s l $5 or d $ 2, and the work done through2 d 2 x B dytyIntegrating in respect of 2,when 2 a l -E, we getWork doneReplacing l by 2 vr y, we getWork done444. Now ty is a function of y , and if f l be the internalpowder pressure, p and R the internal and external radii, andRmpmz — l.substituting which in (23) we get2 2Work doneRdy 1 (24)which gives by integration1 2Work done if(m’fj {m21R2 2 R’ logm445. Proceeding in like manner for compression we getf,’ m2 — 12 a1 R3Work doneE (m’ 2R + 2 R 108 5+and adding these together,Total work doneINTERNAL BALLISTICS. 209446 . If the unities be tons and feet, this gives the workdone per lineal foot, and since the surface of 1 lineal footis 2 or p, making Bunity, the work done per unit of surfaceis got by dividing the surface by 2 71 p, which givesm2 1 f 12(28)Work done per un1t of Surface m2 1 2—EPand this is the work done in foot- tons per square foot ofsurface of chamber.Workdone in Expanding the Chase.447. As the chase varies in thickness, the value of m isnot constant,but the variation is not of such magnitude asseriously to affect the results, and therefore it will be sufficient to assume a mean thickness of the chase,and makeuse of the value ofm belonging thereto .448 . Proceeding thus, the work done per unit of surfacemay be determined as above, using, instead of the constantpressure f l , the varying pressures at each point of the chase,which is a function of the length travelled by the shot.Let L be the total length of travel ;I the equivalent length of the chamber ;a any intermediate length ;P, the pressure in the chamber ;p the pressure at a ;and the work done in d a isfl. m2 1 2 1 74E m’ i l pa P’210 INTERNAL BALLISTIOS.and integrating this, and taking a L,the total workdone is248 1 1(L+Work done in Stretching the Gun between Breech andTrunnions.449. Assuming that the strain is uniformly distributed overthe cross section of the gun, this strain per square inchThen if l be the total length from breech to trunnions,Total extension 41450. For any intermediate extension y, the force an:E if, andE veli yIntegrating and making y the total extension, we getI2 Earea 2 1r (R’and d substituting we getthe work done in dywork done 4“per unit of area ; and since the totalTotal work donehttps://www.forgottenbooks.com/join212 INTERNAL BALLISTI08 .454. Let a be any distance from the point of rest on themuzzle side, and y any distance from the same point on thebreech side , thenVelocity at a u a: (33)Velocity at y a y. (34)455. Let 8 density of products of combustion ;A area of bore.Now the moments are equal on each side of the point ofrest, and these are, on the muzzle side,8A a a,9 land on the breech sideSA a,”2 g u u,’and as these are in opposite directions, their algebraicsum isNow 8A l w, therefore22904456. For the vis viva in the direction of the muzzle wehaveand in the opposite directionINTERNAL BALLISTI08 . 213the integrals of which are8A l us ss z u,”3 g°u +m 3 g u +m’therefore the total vis viva iswhich is the value ofDetermination of R d s,or the Work done in overcoming theresistance of the Air due to Velocity.457. Assuming the resistance to be proportional to thecube of the veloc ity,R a.u“, a being a coeffic ient determinedexperimentally.From experiments made with the Bashforth chronographthe resistance of a 10- inch ogival-headed projectile at 1000feet per second is 233 lbs., therefore the resistance at anyvelocity u is000000233us,a. 000000233.458. Now the veloc ity of the projectile, if Sarrau’s monomial formula be admitted, is C a“, where C is a constant anda the distance travelled by the projectile. Therefore, if u bethe muzzle veloc ity, and l the total travel of the projectilea = C lliC and C214 INTERNAL BALLISTI0S.therefore, the veloc ity atu(m)fira; T ul “ lconsequently-000000233Baa; 000000233 133 G)“and integrating betweena: l and a: 0,°000000233 x 6 ” u3°0000000149 l u3459. This is in lbs. for a 10- inch projectile,and assumingthe resistance to be direc tly as the area, this must be dividedby 144for any calibre c, the resistance will be0000000149 c"~ 554 T;0000000214 c2 l u3to give the resistance per square foot and thereforeor dividing by 2240 for foot-tons0000000000956 02 l 143Unities, feet, seconds, and foot-tons.460. From the equality ofmomenta.0mm= mu +https://www.forgottenbooks.com/join216 INTERNAL BALLISTICS.4215 X 36520,thereforeFall of temperature to tJ AH772 x 6 365 x 300 x 563foot-tons.2240J AQlog {0000191 x 1 f } log 5,Where z is the loss of temperature in degrees Centigrade .r , the time of combustion of the charge, which is hereassumed to be the same as the time of passage ofthe projectile to the muzzle , which is, of course ,a maximum value, and as shown in formula1° 1 488 -51, L and v in metres.L feet metres.v is assumed 2000feet metres per second.1° 0133.mean absolute temperature of products indegrees Centigrade, which by when to is theabsolute temperature of combustion 2274°C., isINTERNAL BALLISTICS. 217ou _ 3 23mu x 2274 x7 295 861 431896°C .a surface in decimetres square 700°3.w weight of charge in kilogrammes 136 °2.Andmaking use of these valueslog 2 log°000191 x x°0133 } log° 72488 ° 711 10From which we get 2 and if c mean spec ificheat ° 186 , and w the we ight of products kilogthe total loss of heat is 23 x 136 2 x° 186 un its(French), or in English units, x 2312, orin the equivalent of work donefoot-tons .A 1 = 0.466 . (a)14 -75 x 78 54 x 22 -5224011 64 foot- tons.(b) Rotation of Projectile.Work done E 11 “y2 g m218 INTERNAL BALLISTI CS.Here W 500lbs.m 30, or one turn in 30calibres.Therefore500 °707 x 22 .T" ) 04255 “or, in foot-tons 000019 na foot- tons.468 .Here byWork doned a;_ 1 737o (zv +which integrated between the limits Land 0gives1 737 p“F, 11337 1° 237mn (L°43 l)’p 5 inches °4166 feet, m 30,P, 2592 tons per sq. foot, l feet, L feet.Making use of which values in the above, we get1 1Work done 298 4°4619 } foot-tons.469. (d) Friction proper of Projectile andBase Ring.The friction of the projectile itself is very small and is by(5440)https://www.forgottenbooks.com/join220 INTERNAL BALLISTI08 .Work done in Stretchmg Gun.471 . By (28) the work done per foot of surface of thechamber iswhereinternal pressure 18 tons per sq. inch ,2592 tons per sq. foot,p 7 inches 05 333 feet,and the surface of the chamber16 5 sq . feet.E modulus of elastic ity13000tons per sq. inch.x 2592 x x2 x 13000 x 144472. By (30) work done is1 1E m2 - 1‘ P1 -474 ( 4 30mun ities, feet, and tons per square foot ; and taking a meanvalue for R the outer radius 8 333.p P, 2592,E 13000X 144,l = 8 ° 232,L = 22 °5,I 8 2 2 lm + 1 B + p 8333 4166INTERNAL BALLISTICS. 221ThereforeI 17 425 foot- tons.W = ao-75 {-1546Stretching between Breech and Trunnions.473. By (32) work done isHenceR 20 ft. °4166 ft. 1 9 ft.aR‘p’Ix x 9 x 2592”13000 X 144 x 152 1 ° 173foot-tons.Determination of A V.(1) Vis viva of projectileW9Work doneW 500 lbs.° 2232Work done2 x 32 . 2003465 u foot-tons .475. (2) Vis viva of gun and recoiling part of carriageW“:9222 INTERNAL BALLISTI08 .Taking the weight of gun and recoiling part of carriage at36 tons,Work done 559 n,“476. (3) Vis viva of gases, orBy (42) this is“I?and work done (n’n,“w 1339 tons.1339a 3 _Work done6 x + u,°000693 (n2n2n ul ) foot-tons.Resistance of Air R dw.477. By (44) this is °0000000000956 c"’l n3Hence8333 foot.The value of u must be assumed, and in this case 'we willtake it at 2000 feet per second, which gives work done1 1 45 foot-tons.https://www.forgottenbooks.com/join224 INTERNAL BALLISTI08 .substituting which in the above we get16747 004207 a”,u 1995 feet per second,which is the muzzle ve loc ity ; and°00805 1995 feet per second ;which is the velocity of reco il.479. From which we getWork done for rotation 000019 a”75 62 foot- tons.friction of gun 1940 00on projectile °003465 a” 14265 00on gun and carriage 559 n,a 149 °00on gases°000693 (u2a2l ug} 2743 00480. Summary of Work done .13796 0foot- tons,gun and carriage in recoil 149 °0a: 83599Friction of gasesRotationResistance of air (statical) 1 1 641 1 45Friction of projectile 1 01in riding 83 34Stretching gunEquivalent of heat imparted to gunTotal 19734 ° 7 foot- tons.INTERNAL BALLISTI08 . 225481 . The equivalent of the heat expended, as shown inwas foot-tons, so that the diffe renc e is154 foot-tons, or 0° 75 per c ent.of thenitroglycerine has increased the force of the black powder,and from this it has been concluded, that for black powdertwo orders of explosion may exist. The first order be ingthat of detonation ; the second, that of ordinary combustionand it is stated by French authorities that all explosions aresusceptible of these two orders of explosion according to thec ircumstances under which they are fired.Not that an absolute distinction can be shown in eachparticular case, but that those orders are the limits betweenwhich practical results may take place.In this way the force of black powder may vary from1 to 3°3.Captain Roulin, of the French Artillery, gives the followingtable of results showing the relative force of the two orders ofexplosion in different explosives.INTERNAL BALLISTICS. 5Explosive . l st order. 2nd order.Black powder10° 13GuncottonPicrate of potassaFulminate ofmercuryThe order of the explosion which will take place dependsupon external conditions offiring.Dynamite and guncotton may be burnt in the open airwithout any explosion by the simfle contact of flame ; onthe other hand, if the gases are confined by an envelope ofmore or less resistance, or if thematerial be previously heatedto a certain degree, there will be an explosion of the 2ndorder ofmore or less violence .If,however, the explosion is effected by means of adetonator, such as fulminate of mercury, this will give riseto a detonation or explosion of the first order.7. It is evident, therefore , that the pressure existing in thebarrel of a gun with any particular powder may be greatlyaffected by the c ircumstances under which the ignition takesplace, and the subject has much interest as regards the socalled wave pressures in guns.M. Berthelot treats of this subject in his treatise Sur 1aforce des matieres explosives d’apres la Thermochimie,’3rd cd., 1883.According to his views every explosive reaction must bereferred to an initial rise of temperature, whichmay be causedby ignition,by a shock, or by friction , and this reaction istransmitted from particle to particle successively. Certainexplosive materials decompose spontaneously and slowlyunder ordinary temperatures, without explosion , whilst theirdetonation takes place when the temperature is suddenlyand largely increased either purposely or by acc ident.6 INTERNAL BALLI8 TI08 .M. Berthelot distinguishes the propagation of the reactionby two c lassesl st, That of combustion.2nd, That of detonation .Between these two classes there may exist a series ofintermediate modes of explosion. In fact, the passage fromone class to another is accompanied by violent and irregularmovements of the material, during which the propagation ofthe combustion acts by a vibratory movement of increasingamplitude and with more or less velovitv.Class of Combustion .8 . When an explosive material is gradually heated toa sufficient degree, a portion of it explodes ; if the gasesare free to expand, the pressure rises slowly and a freshportion of the material is ignited, and thus the inflammationis propagated from particle to particle with a veloc itydependent on the circumstances of the case . Such is generally the course of action with ordinary gunpowder.Class of Detonation .9. When a shock sufficiently violent is produced in onepart of an explosive substance , and if the pressures whichresult from this shock are too sudden to be propagated tothe whole mass, the transformation of the vis viva into heatwill take place chiefly in the first portion of the mass.This may thus be raised to a sufli c ient temperature todetonate . If the first production of gas is so rapid that themass of the material has not time to be displaced, and if theeXpanswn of the gas produces a more and more v iolentshock on the adjoining portion of the material , the v is vivaof this new shock will be transformed into heat, and thusgive rise to the detonation of a new portion of the material .This alternate action of a shock the vis viva of which istransformed into heat, and a produc tion of heat which raisesthe temperature of the next portion so as to produce a newINTERNAL BALLISTICS. 7detonation,transmits the reaction from portion to portionthroughout the entire mass.The propagation of the inflammation then in this c lass ofdetonation may be compared to that of a wave of sound,that is to say, it is a true wave of explosion travelling with aveloc ity incomparably greater than that of a simple ignition transmitted by contact from partic le to particle, andwhen the gases freely expand as they are produced. Itmust also be remarked that whilst the wave of sound isgenerated by a periodic succession of similar waves, that ofexplosion is not periodic , but takes place once for all .10. An Sxplosion of the second order may be transformedinto one of the first.The velocity of propagation of reaction in a case of thesecond order, is greater as the molecular intensity of reaction is greater, this being defined by the quantity ofmaterialtransformed into gas at a fixed temperature and underconstant pressure .It increases also, (1) with an increase of the initial temperature of the mass.(2) With the increase of the weight of the charge,because in this case the influence of cooling is proportionatelyless.(3) With the inc rease of pressure under which the gas isgenerated.When the explosive matter is confined by a tamping, thepressure will rise very rapidly, and the veloc ity of propagation may be sufficiently great to give rise to a pressure or ashock capable of detonating a portion of the mass.This is no doubt the case in long charges of small grainedpowder ignitedat the rear. The forwardportion ofthe chargeis jammed up against the projectile , and the reaction is converted from one of the second to one of the first order,giving rise to the so-called local wave pressures whichhave been observed.In operating with an explosive of which the molecularveloc ity of reaction is very great, such, for instance, as nitro8 INTERNAL BALLISTICS.g lycerine or fulminate ofmercury,no tamping is required ;the gases are developed so rapidly that the environment, beit solid, liquid, or even gaseous, has not time to be displaced,and opposes itself like a fixed wall to the action of theexplosive during the infin itesimally small time of reaction.Of the Potential of an Exp losive.11 . By the“Potential of an explosive ismeant themechanical equivalent of the heat given out by its combustion.Thus if E be the mechanical equivalent of heat,and Qthe un its of heat developed by the combustion of unit ofweight, EQ is the “ Potential .” This Potential is independent of the manner in which the combustion takes place,provided that it be complete and the final state of theproducts the same .Making use of French unities, and the French equivalentof heat, or E 436, Messrs. Sarrau and Roux determinedthe Potentials of several explosives as given in the followingtablePOTENTIALS or EXPLOSIVES.Composition.Denominations . £2321tFrench .Fine sporting powderOrdinary common powderRifle powder, BM ining powderNitroglycerineGuncottonPicrate of potassaNoble and Abel .CocoaSpan ish pellet 75 6 12 5 1 1 ° 5Curtis and Harvev . No . 6 71 °7 10° 4 14°0FHG . WalthamAbbey 74 ° 10° 1 15 °E.L G. 10° 1 14 °3Pebble 10° 1 l 4 ° 2Mining oz 15 2 21 4INTERNAL BALLISTICS. 912. The Potential of an explosive must not be confoundedwith the mechanical effect which may be obtained from it ;neither must it be confounded with the pressure developedby exploding it in a close vessel.In treating hereafter of the action of gunpowder thesedifferences will be fully explained.13. As regards explosives, a distinction may be madebetween two classes.First,mechanical explosives, in which the reagents areuncombined,but intimately482 . Taking the whole heat developed in the combustiono f the powder at 1298 °4 units per 1h the equivalent work of300 lbs. is . foot-tons.Of this there is accounted forshowing a loss of foot-tons ; the wholeof which is in the residual heat of the gases as they escapefrom the gun at an absolute temperature of 3652’ Fab.483. It thus appears, that if we consider the energyimparted to the projectile, the useful effect only13795 1134260is utilised.484. Of the work actually done in the gun, the followingare the percentageslbs., or about 103 per cent. of the powerUseful , efl’ect on projectileEffect on recoilon the gasesOn rotation and friction of pro]ectilesOn expulsion of air stretching gunBeating the gun100 000485 . The observed veloc ity with this gun was about 2100feet per second, but as the observed veloc ity is always somewhat greater than the real muzzle veloc ity, owing to thecontinued action of the gases, the actual muzzle veloc itywould be about 2065 feet per second, or about 3§ per cent.above the calculations. This difference is not more thanmight be expected, since the two important items of loss ofheat and friction of gases are subject to considerable uncertainty . It appears, therefore, that the Thermodynamicmethod gives very approximately correct results.48 6. The foundation of it is of course the fall of temperaQper cent.76424 240° 810250226 INTERNAL BALLISTICS.ture in the gun, or more correctly stated, the differencebetween the temperature of combustion and the temperatureof the products at the time the projectile reaches the muzzle ,and this is got by using Noble andAbel’s equation, wherethe change of temperature is given as a function of thechange of volume .487. In this equation it is supposed that the whole of thepowder is converted into products before the change ofvolume begins, and that it is then at the highest temperature, and that afterwards in expanding and doing work, itfalls to the lower temperature .488. To this it may be objected, that in the case of a gun,these conditions do not exist that the powder begins to dowork as soon as the combustion begins ; that it also beginsby imparting some part of its heat to the gun, consequentlylowering the temperature and pressure of the products, andit may be said, that under such a simultaneous generationand expenditure of heat, the result must be very differentfrom that of the first hypothesis. But those who hold thisview ought not simply to assert that it may be so, but toshow why itmust.489. The application above given of the Thermodynamicmethod to the l O- inch gun shows that it gives results whichpractically agree with experiment, and the inference is,that although the process of combustion may vary, the effectis practically the same.490. This is indeed distinctly stated by Count de St.Robert,who says Princ ipes de Thermodynamique ,’ p. 2512,Turin,Quel que soit la mode de combustion de lacharge de poudre dans l’arme 3 feu, qu’elle se consumeinstantanément ou successivement, les deux temperaturest t seront toujours les memes. La premi‘ere dépend dela composition de la poudre ; elle est déterminée par lareaction chimique qui s’op‘ere pendant que la poudre passe31 l’etat gazeux. La seconde no dépend que du rapport del’espace , occupé par les gaz lorsqu’ils ont la temperature to al’espace qu’ils occupent apr‘es la détente dans l’ame derrierehttps://www.forgottenbooks.com/join228 INTERNAL BALLISTI08 .CHAPTER VI.Concluding Remarks.493. Up to the present time powder composed of nitrateof potassa, charcoal, and sulphur has been almost exclusive lyused for artillery, and the proportions of these ingredientshave not variedmuch, but a very great alteration has takenp lace in the size and form of the grains. These latter havevery largely inc reased, with the view of reduc ing the pressure , and this object has been still further attained by thePrismatic form, whereby the Ignition is made more gradual .494. Whilst, in this way, the pressure has become reduced,it has been necessary largely to inc rease the charges rela~tively to the weight of projectile, and this , again, has necessitated the increased length of guns. It has further led toinc reased erosion and other inconvenienc es.495. It may very well be asked whether too much has notbeen sacrificed to this reduction of Pressure,andwhether ourOrdnance Department has been,and is at the present time ,on the right track in gun construction .496. It may be asked, Whe re is the necessity of limitingthe maximum pressure in"a gun to 16 or 17 tons per squareinch ?497. In a gunmade solid, and of homogeneousmetal, thereis indeed a limitation to the internal pressure that it willbear, viz. the tensile force of the metal ; but in a built-upgun this limitation is not necessary, and it is quite possibletomake a gun which will not be strained to more than 20 tons,whilst the internal pressure may be 40tons per square inch .If, then, the elastic limit of the material of thi s gun be25 tons, it is perfectly certain that it may be subjected to aninternal pressure of 40 tons without injury. Why, then,INTERNAL BALLISTI08 . 229limit the internal pressure to 17 tons, thereby involving alarger charge andmany other inconveniences498 . It is perfectly certain, as appears from M. Sarrau’sinvestigations, fully confirmed by experimental results, thatthe ballistic effec t of a given weight of powder increases asthe maximum pressure inc rehses, and therefore the object ofthe gun constructor shouldbe to increase the strength of thegun, so as to master the force of the powder, rather than toseek for a weak powder to suit a weak gun, and increase theballistic effect by an increased we ight of charge,and lengthof gun.This is the path entered on a few years ago, and stillpersisted in by our gun manufacturers, and it is by thisprinc iple that our Ordnance Department is now guided, andour new armament is be ing construc ted.499 . Nothing has been said in the preceding chaptersabout the new powders which have been invented in Franceand in this country, because everything connectedwith themis kept a profound secret, excepting that from time to timewe are startled with results said to be obtained with thesenew explosives.500. On a recent occasion Lord Armstrong, presiding ata meeting of the Elswick Company, is reported to have saidthat with the powder nowmade by the Chilworth Company,a charge of one-third less weight than hitherto used gives amuzzle veloc ity of02400 feet per second that is to say,a higher velocity than the ordinary powder. He saidnothingabout the maximum pressure, but as this is limited by theOrdnance Department to 17 tons per square inch, it is to bepresumed that this was not exceeded. Now, it is certain thatno such effect can be produced with a charcoal and nitratepowder. The force, or strength, of a powder is, as shown inP0”0To108) denoted by 273all such powders ; but it is‘quite possible that the product ofthe two variables v0 and To may be increased by the use ofother ingredients. If, for instance, some ingredient otherwhich is nearly constant for230 INTERNAL BALLISTI08 .than charcoal were used, capable of giving an increasedvolume of gas with the same value of To, a more powerfulpowder would result. If, again, no remains constant, whilstT0 was increased, the same result would ensue, andafortiori,if both so and To were increased, a stillmore powerful powderwould be obtained.501 . The question,however, still remains,What will be theeffect of such powder as regards the maximum pressure , theerosion of the gun, and the storage and keeping properties ofsuch powder502. Stability of constitution is of the utmost importance in gunpowder. If it be liable to change so as toalter its characteristics,” it is evidentthat range tablesmust become of little use .503. Such change may be brought about either by achange in the hygroscopic or in the chemical condition ofthe powder.504. Charcoal powders are only subject to the first ofthese changes, and the less moisture they contain the morestable they are . For this reason it may be expected thatcocoa powder, which contains a high percentage of moisture,will be more subject to change and become more violent inhot climates than the black powders, which contain lessmoisture .505. Charcoal powders are not subject to chemical changeexcept at very high temperatures. On the other hand, powderscomposed of substances which have greater chemical aflinityinter so, may be expected to undergo considerable changewhen kept under certain conditions differing from thoseunder which they were manufactured.506. Although the composition of the new powders is keptsecret, it is very probable that they are to a great extentcompounds of ammonia and picrates or their analogues, andif so they are no doubt liable to change ifkept for a time in amoderately high temperature, such as would obtain on boardship or in magazines in the tropics.507. Such change might not only affect the chemical conhttps://www.forgottenbooks.com/join232 INTERNAL BALLISTI CS.POSTSCRIPT.AFTER the preceding pages had gone to press, I received,through the kindness of my friend Lieut. Crozier, copies ofNotes on the Construction of Ordnance ,’ Nos. 36 and 42 .No. 36 is by Lieut.W . M. Medcalfe , of the Ordnance Department U.S. Army, and is dated 28 th April, 1886 . It isentitled Application of Sarrau’s Formulas to AmericanPowders and Guns.’No. 42 is a translation of Sarrau’s Recherches Théoriquessur lo Chargement des Bouches é feu ,’with Notes andAppendix by Lieut. D . A. Howard,Ordnance DepartmentU.S. Army, and is dated 19th August, 1887.The publication of these documents for the use of oficersof the Ordnance Department, by authority of the Secretaryof State for War of the United States, is evidence of thehigh value attached in that country to M. Sarrau’s investigations , the practical value of which is evident by the tableswhich are reproduced at the end of this Postscript,andwhich will be found of great interest to artillery officers.Table 1 . c ontains the ballistic elements of various Americanservice guns, with the ir relative powders and the ballisticresults obtained.Table 11 . gives the characteristics of sixteen Americanpowders.Table III. shows the verification of the characteristicsby comparison of the actual muzzle relations and pressureswith those deduced from the formulae.Table A,taken from Lieut. Howard’s Appendix toNo . 42, gives the “ Characteristics of a number of brownprismatic powders tested in an 8 - inch gun .INTERNAL BALLISTI08 . 233Table B. The verification of some of these same powdersin the same gun .An examination of these tables, which give the results of138 rounds fired from guns varying from -inch to 12- inchcalibre,and with different powders, shows how satisfactorilyM. Sarrau’s method represents the actual facts of artillerypractice, and consequently how important his investigationsare, both as regards ballistic practice and the constructionof guns.Lieut. Medcalfe, in paperNo. 36, points out that the valueof the force of the powder is not constant, espec ially asregards the brown prismatic powders, and in the Table II .it will be seen that it varies from to 0 735 in cocoapowder.The probability of this variation is pointed out in (5327)page 153 ante,and the value off for cocoa powder was thereestimated at 0 7635 .The value off for cocoa powder must, however, be takenwith much reserve. It is well known that with this powder,and especially when fired in large charges, some of the grainsare blown out only partially consumed, and of such a caseM. Sarrau’s formula does not take account ; and the valueof f determined from any particular experiment must beconsidered as only an approximate value, and correct only asregards the ballistic elements of that experiment.It should be the object of the artillerist to have such avalue of 'r as will ensure the complete combustion of everygrain before the projectile leaves the gun.Lieut. Medcalfe observes in Notes 36, page 10, thatAny change in the method of manufacture which favoursthe regularity of burning of the grains will inc rease theeffective strength, and enable us to obtain with the samemaximum pressure higher veloc ities.“Assuming f 1 as a mean value of the powder that iseasily obtainable, and giving for ordinary densities andpressures values of 7°small enough to obtain the completecombustion of the grains before the projectile leaves the gun,234 INTERNAL BALLISTI08 .we will attempt to deduce the relation between the densityand 'r for brown prismatic powder.NV1 , Table 11. may be taken as the standard, and thevalue ofK inM . Sarrau’s equation 7 K (1 0 875 8) 243)page 119 ante,makinge inches 0 121 dm.8 l ‘ 828KLieut. Medcalfe shows, No . 36, page 9, that with thel 2- inch gun the best result is obtained with a charge of246 lb. of powder, and a value of 7 1 5 93 (see 307,page 144 ante, and No . 36, page and using the value ofK 1 2 17 just obtained, we find 8 1 ° 804, which is nearlythat ofN.R.,Table 11.Referring to M. Sarrau’s formula (15) 174) and (42a)Lieut. Medcalfe observes that with slow powdersft“2f artsequently 1ff and a remain constant, any increase of 7' whichreduces the pressure must also reduce the veloc ity, thoughin a less proportion. If, however, we wish to decrease thepressure and retain the veloc ity unchanged,f and 7 must befboth increased in such a manner that2;shall decrease, whilstthe veloc ity varies as and the pressure as and conremains constant.Taking, for instanceCocoa powder 7 1 130andf ‘ 735, andN.V. powder7 _ 1 952 and f 968, we get’g 8 32 for both, and6 50 for cocoa, and °496 for N.V. Consequently,https://www.forgottenbooks.com/joinINTERNAL BALLISTICS.236cooma8mamacascomumcamcmcan3commm5208.{H.230mm”8taSewn”95605»4a.moutd.mINTERNAL BALLISTI08 . 237O Q O Q Q3 6 3,3 3 2 3 35 5 5355 5 5 5 5m0 2.INTERNAL BALLISTICS.238an”mgmgIDmooN N N N N N g Ndd2885canddgoes.”asasec86aread5https://www.forgottenbooks.com/joinINTERNAL BALLISTIO'S'.awO(NIn3533882 3838smooomoooonmumomoomomoonomoononownemon~wmoowomonoNooow«mooh_mownhmmum«mowommooho¢oooonwoommoovonNmobmobohmbob«muNowhobamo-wowwwanmummommumwwwmvwabwfiestamowABBREEQQQXw34 4 4 4mmo y o dmmmmmmma d o o o o2:p,ii( 5LONDON:PRINTED BY W ILLIAM CLOWES AND SONS, LIMITED,STAMFORD STREET AND CHABING CROSS.ORDNANCEA TREATISEONTHEAPPLI CATIONOFWI RETOTHECONSTRUCTIONOFORDNANCE.By JAMES ATKINSON LONGRIDGE, Mem. Inst. (LE.8 vc , c loth , £ 1 5 8 .The Treatise before us introduces to the public for the first time, theresults of investigations with our guns, commenced as long ago as 1855,which are continued up to the presentdate . The work must be of unusualinterest to all those who care for the future prosperity of England, whichis wrapped up so closely with the question treated of in this work.”Jackson’s Woolwich Journal , August l st, 1884.We think Mr. Longridge has made out hi s case, and that his systemdeserves a fair trial in comparison with other promising systems.”— Nature,July 24th , 1884 .E. a F . N . SPON , 125 , STRAND ,LONDON .NEW YORK :12 , CORTLANDT STREET.https://www.forgottenbooks.com/joinCATALOGUE OF SCIENT IFIC BOOKS7 14“ Publi shed, in D ene] 8m, cloth, con ta ining 975pages and 2 50 I llustra tions , pr ice Gel .SPONS’HOUSEHOLD MANUALA Treasury of Domestic Receipts and Guide for Home Management.PRINC IPAL CONTENT S .sel ecting a.od H ou s e , pointing out the essential requirements fora good house as to the Site, 801 Trees, Aspe c t. 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N. SPON. 3T h e Bath -Ro om.—Bathing in General ; Management of Hot-Water System.Th e Laundry —Small D omestic Wash ing Machines, andmethods of getting up linenFitting up andWorking a SteamLaundry .Th e Sch ool -Boom.—The Roomand its Fittings ; Teaching, etc.Th e P lay g round —Air and Exercise ; T raining ; Outdoor Games and Sports.Th e W orkro om.—Darning, Patch ing. and M ending Garments.Th e Library —Care of Books.Gfi'gh eG arden —Calendar of Operations for“Lawn, Flower Garden, and Kitchenen.Th e Farmy ard—Management of the Horse , Cow, Pig, Poultry , Bees, etc . , etc .Sma ll M otor 8 —A description of the various small Engines useful for domesticurpooses, from r man to 1 horse power, worked by various methods. such as Elec tricugines , Gas Engines, Pe troleum Engines, Steam Engines, Condensin Engines, WaterPower, Wind Power, and the various methods of working and managing t em.H ou seh o ld Lew .—The Law relating to Landlords and Tenants, Lodgers, Servants,Paroch ialAuthorities, Juries , Insurance, Nuisance, etc.On D esigning Belt Gearing . By E. J. COWLINGWELCH, Mem. Inst. Mech. Engineers, Author of Designing ValveGearing.’Fcap. 8vo, sewed, 6d.A H andhooh of Formula ,Tahles, ana’ M emorana'a,for Architectural Survey ors and others engaged in Building. By J. T .H URST , C .E. Fourteenth edition, royal 32mo , roan, sr.It is no disparagement to the many excellent publications we refer to, to say that in ouro p inion this little pocket-hook of Hurst’s is the v ery be st of themall, without any excep tion.It would be use less to attempt a recapitulation of the contents, for it a pears to contain almostevery/thin that anyone connec ted w i th building could require and, est of all, made up in acom ac t armfor carry ing in the pocket, measuring only 5 in . b 3in ., and about i in . thick,in a imp cover. We con ratulate the author on the success 0 h is laborious and practicallycomp iled little book, whic has received un ualified and deserved praise fromevery profess ional person to whomwe have shown it.”he D ublin Builder .Tahulatea’ Weights of Angle, Tee, Bulh, Rouna',Square, and Flat Iron and Steel, and o ther information for the use ofNaval Architects and Shipbuilders. By C . H . JORDAN,Fourthedition, 32mo , cloth, as. 6d.A Complete Set of Contract D ocumentsfor a CountryLodge, comprisin D rawings, Spec ifications, D imensions (for quantities),Abstracts, Bill ofgQuantities, Form of T ender and Contrac t, with No tesby J. LEANING, printed in facsimile of the original documents, on singlesheets fcap ., in paper case, i os.A Practical Treatise on H eat, as applied to the(Awful Arts ; for the Use of Engineers, Architects, &c . Bév THOMASBox. "67th I4 plates. Third edition, crown 8vo , cloth, 1zs. cl.A D escriptive Treatise on M athematical D rawingInstruments:their construction, uses, qualities, selection, preservation,and suggestions for improvements, with hints upon D rawing and Colouring. ByW . F. STANLEY,M .R.I. Fifth edition, with numerous illustrations,crown 8vo, cloth, sr.4 CATALOGUE OF SCIENTIFIC BOOKSQuantity Survey ing . By J. LEANING. W ith 42 illustrations. Second edition, revised, crown 8vo, cloth , 9s.CONTENTSA complete Explanation of the London Schedule of Prices .Practi ce . Formof Schedule ofPrices.General Instructions. Analysis of Schedule of Prices .Order of Taking Oll'. Adjustment ofAccounts.M odes ofMeasurementof the variousTrades. Form of a Bill ofVariations.Use andWaste. Remarks on Spec ifications .Ventilation andWarming. Pri ces and Valuation of B'ork, w ithCredits, with various Examp les ofTreatment. Examples and Remarks upon each T rade .Abbreviations. The Law as it afi'ectsQuantity Surveyors,naring the D imensions . wi th Law Reports .A tracting , with Examples in illustration of T aking OKafter the oldMethod.each Trade. Northen Prac tice.Billing . The General Statement of the MethodsExamples of Preambles to each Trade . recommended“by the Manchester SocietyFormfor a Bill of uantities. ofArchitects for takingQuan tities.D o . Bill of redits . Examples of Collec tions .D o. Bill for Alternative Estimate. Examples of T aking OK” in each T rade .Re torations and Repairs. and Form of Bill. Remarks on the Past and Present MethodsVariations before Acceptance of T ender. of Estimating.Errors in a Builder’s Estimate .Spons’A rchitects’ana'Builders’P rice Booh, withuse ul M emoranda. Edited byW . YOUNG, Architect. Crown 8vo , cloth ,red edges, 3s . 6d. Published annually . Sixteenth edition. New ready .Long- Span Rai lway Bmdges, comprising Inv estigations of the Comparative Theoretical and Practical Advantages of thevarious adopted or proposed Type Systems of Construction, w ith numerousFormulae and Tables giving the weight of Iron or Steel required inBridges from300 feet to the limiting Spans ; to which are added similarInvestigations and Tables relating to Short-span Railway Bridges. Secondand revised edition. By B. BAKER, Assoc . Inst. C .E. Plates, crown 8vo ,c loth, 5s.Elementary Theory and Calculation of Iron Bridgesand Roe/fir. By AUGUST RITTER, Ph .D . , Professor at the PolytechnicSchool at Aix-la-Chapelle. Translated from the third German edition,by H . R. SANKEY, Capt. R.E. W ith 500 illustrations, 8vo , cloth, 153.The Elementary P rinciples of Cargentry ByTHOMAS TREDGOLD . Revised from the original edition, and partlyt e -written, by JOHN THOMAS HURST . Contained in 51 7 pages of letterpress, and i llustrated w ith 48 p lates and l 50 wood engrav ings. Sixthedition, reprinted from the third, crown 8vo , cloth , 1 2s . 6d.Section I . On the Equality and D istribution of Forces Section II. Resistance ofT imber— Section III. Construc tion of Floors— Section IV. Construc tion of Roofs— Section V. Construc tion of Domes and Cupolas— Section VI. Construction of PartitionsSection VII. Scafl'o lds, Staging, and Gantries—Section VIII. Construc tion of Centre s forBrid es—Section IX. Cofi'er-dOams, Sharing, and Strutting— Sec tion X . Wooden Bridgesand inducts—Section XI. Jomts, Straps, and other Fas tenings- Sec tion XII. T imber.https://www.forgottenbooks.com/join6 CATALOGUE OF SCIENT IFIC BOOKSA P ractical Treatise on Coal M ining By GEORGEG . ANDRE, F.G .S., Assoc. Inst. C .E. ,M ember of the Society ofEngineers.PVzth 82 lithographic plates. 2 vols , royal 4to , cloth , 31. 1 2s.A Practi cal Treatzse on Casting and Founding ,including descriptions of the modern machinery em loyed in the art.N. E. SPRETSON, Engineer. Third edition, wit 82 p late:drawn110scale, 41 2 pp . demy 8vo, cloth, 18r.The D epreciation of Factor ies and their Valuation .By EW ING MATHESON,M Inst. C .E. 8vo, cloth , 6s.A H andhooh of Electrzcal Testing . By H . R. KEMPE,Fourth edition, revised and enlarged, crown 8vo , cloth , 16s.Gas Worhs:the ir Arrangement, Construction, Plant,and Machinery . By F. COLYER, M . Inst. C .E. With 3!folding p lates,8vo, cloth , 24s .The Clerh o/Worhs:a Vade -M ecum for all engagedin the Superintendence of Building Operations. By G. G. HOSKINS,F.R.I.B.A.pThird edition, fcap . 8vo , cloth, Is. 6d.Amer ican Foundry P ractice T reating of Loam,Dry Sand, and Green SandM oulding , and containing a Practical T reatiseupon the Management Of Cupolas, and the M elting o f Iron. By T . D .WEST , Practical Iron Moulder and Foundry Foreman. Second edition,with numer ous illustrations, crown 8vo, cloth, 1 0s . 6d.The M aintenance of M acadamised Roads . By T .CODRINGTON, General Superintendent ofCounty Roadsfor South Wales. 8vo , cloth , 6s.Hydraulic Steam and H and Power L if ting andPressing Machinery . By FREDERICK COLYER, M . Inst. C .E.,M . Inst.M .E.PVith 73p lates, 8y o , cloth , 1 8s.Pumps and Pumping M achinery . By F . COLYER,M .I.C.E. , M . .I M .E. PVzth 23folding p1ates, 8y o , cloth , 1zs. 6d.Pumps and Pumping M achinery By F . COLYER.Second Part. W'i th I I largeplates, 8vo, cloth , 1 2 s. 6a'A Treatise on the Orig in , Progress , P revention, andCure of D ry Rot in Timber , with Remarks on the Means of PreservingW ood fromD estruction by Sea-W orms, Beetles, Ants, etc . By THOMASALLEN BRITTON, late Surveyor to the M etropol itan Board of Works,etc., etc . Wi th IO plates, crown 8vo , cloth, 7s . 6d.PUBLISHED BY E. F. N. SPON.The M unicipal and Sanitary Eng ineer’s H andhooh.By H . PERCY BOULNOIS, Mem. Inst. C .E., Borough Engineer, Portsmouth . PVith numerous illustrations, demy 8vo, cloth, 1zs. 6d.CONTENTSThe Appointment and Duties of the T own Surve or—Traflih MacadamisedRoadway sSteamRolling- r Road ‘Metal and Breaking—Pitched avements—Asphalte—WoodPavements—F ths— Kerbs and Gutters—Street Naming and Numbering— Street Lighting—Sewere entilation of Sewers—D isposal of Sewage—House D e—D isinfec tion—Gas andater Companies, etc. , Breaking up Stree ts— 1m rovement of°vate Streets—BorrowingPowers—Artizans’and Labourers ’ Dwellings—Pub ic Conveniences—Scavenging, includingStre et Cleansing—Watering and the Removing of Snow—Planting Street T rees—Deposit o i'Plans—D an erous Buildings—Hoardings—Obstruc tions—Improvm Street Lines— CellarO nings ublic Pleasure Grounds—Cemeteries—M ortuaries— Catt e andOrdinary Marketsublic Slaugh ter-houses, etc —Giving numerous Forms of Notices, Specifications, . andGeneral Information upon these and Other subjects of great importance to Municipal Engineers and others engaged in Sanitary Work.M etrical Tahles . By G . L. MOLESWORTH,32mo, cloth, rs. 6d.CONTENTS.General—Linear Measures—Square Measures—Cubic Measures—Measures of CapacityWeights—Combinations—Thermometers.Elements of Construction f or Electro-M agnets. ByCount TH . DU MONCEL, Mem. de l’Institut de France. Translated fromthe French by C . J . WHARTON. Crown 8vo, cloth , 4s. 6d.Practical Electrical Units Popularly Exp lained, w ithnumerous illustrations and Remarks. By JAMES SwrNBURNE, late ofW . Swan and Co ., Paris, late ofBrush -Swan Electric LightCompany,.S.A. 18mo, cloth,A Treatise on the Use of Belting for the Transmission of Power . ByJ. H. Coopn . Second edition, illustrated, 8vo,cloth, 1ss.A Pochet-Booh of Useful Formulce and M emorandafor Civil and M echanical Engineers. By GUILFORD L. MOLESWORTH,Mem. Inst. C .E. ,Consulting Engineer to the Government of India forState Railways. With numerous illustrations, 744 pp . Twenty -secondedition, revised and enlarged, 32mo , roan, 6s.SYNOPSIS or CONTENTSSurveying, Levelling, etc .- Strength and Weight of Materials—Earthwork. BrickworkMasonry , Arches, e tc—Struts, Columns, Beams, and T russu—Flooring, Roofing, and RoofT russes—Girders , Brid es, e tc .—Railways andRoads—Hydraul ic Formula—Canals. Sewers,Waterworks, D ocks rrigation and Breakwaters—Gas, Ventilation, and Warmin - Heat,Li ht, Colour, and Sound—Gravi Centres, Forces, and Powers—Millwork, eeth ofee ls, Shafting, eta- Workshop ec ipes—Sundry Machinery—Animal Power—Steamandthe Steam Engme—Water—power, Water-wheels , T urbines, etc .—Wind and WindmillsSteam Navigation, Sh ip Building, Tonnage, e tc .—Gunne Projec tiles, eta—We ights,Measures , and Money—Trigonometry , Conic Sections. and w ee—Telegraphy—Mensuraption—Tables of Areas and Circumference , and Arcs of Circles°thms, Square andCube Roo ts, Powers—Remprocals , etc —Useful Numbers—D ifl'eren and Integral Calculus—Algebraic Signs—T elegraphi c Construction and Formula .CATALOGUE OF SCIENTIFIC BOOKSH ints on A rchitectural D raughtsmanship. By G .W .TUXFORD HALLATT. Fcap. 8vo , clo th, rs. 6d.Spons’Tahles and M emoranda f or Engineersselected and arranged by J. T . H ORST , C .E., Author of Architec turalSurveyors’ Handbook,’Hurst’s Tredgo ld’s Carpentry,’etc. Ninthedition, 64mo , roan, gilt edges, rs. or in cloth case , 1s. 6d.This work is printed in'agear! type , and is so small, measuring only a} in . by r!in. byt in. thick, that itmay be eas y earn ed in the waistcoat pocket.It is certainly an extremely rare thing for a rev iewer to be called upon to notice a volumemeasurin but a} in. by t ! in. , t these dimensions faithfully re sent the size of the handylittle boo before us. The vo ume—which contains 1 1 8 prin pages, besides a few blanpages formemoranda—is, in fact, a true pocket-book, adapted for being carried in the waistcoat pocket, and contaimng a far greater amount and variety of information than most peoplewould imagine could be compressed into so small a spac e. The little volume has beenc omp iled w ith considerable care and judgment, and we can cordially recommend it to ourreaders as a useful little pocket companion.—EW m .A Practical Treatise on N atural and ArtificialConcrete, its Varieties and Constructive Adaptations. By HENRY REID ,Author of the Science andArt of the Manufacture of Portland Cement.’New Edition, with 59 woodcuts and 5p lates , 8vo, c loth, 15s.N otes on Concrete and Worhs in Concrete ; espec iallyw ritten to assist those engaged upon Public Works . By JOHN NEWMAN,Assoc . Mem. Inst. C .E., crown 8vo , c loth, 4s. 6d.Electricity as a M otive Power . By Count TH . DUMONCEL, Membre de l’Institut de France , and FRANK GERALDY , I 6nieur des Ponts et Chaussées. Translated and Edited, w ith Additions, yC . J. WHARTON, Assoc. Soc . T el. Eng. and Elec. M t}:1and diagrams, crown 8vO, c loth, 7s. 6d.Treatise on Valve Gears, w ith spec ial considerationof the Link-Motions ofLocomotive Engines. By D r. GUSTAV Z EUNER,Professor of A plied Mechanics at the Confederated Polytechnikum ofZ urich. Trans ated from the Fourth German Edition, by Professo r J. F.KLEIN, Lehigh University, Bethlehem, Pa. Illu strated, 8vo , clo th , 1 2s. 6d.The French Polisher’s M anual. By a FrenchPolisher; containing Timber Staining, Washing, Matching, Improv ing,Painting, Imitations, Directions for Staining, Sizing, Embodying,Smoo thing, Spirit Varnishing, French -Polishing, Directions for Repolishing. Third edition, royal 32mo, sewed, 6d.H ops, their Cultivation, Commerce, and Uses invarious Countries. By P. L. SIMMONDS. Crown 8vo, cloth, 4s . 6d.The Principles of Graphic Statics. By GEORGESvDENHAM CLARKE, Capt. Royal Engineers. With 1 1 2 illustrations.4to , cloth, 1 2s. 6d.https://www.forgottenbooks.com/joint o CATALOGUE OF SCIENTIFIC BOOKSPerspective, Explained and Illustrated. By G . S.CLARKE, Capt. R.E. W'ith illu strations, 8vo , c loth, 3s. 6d.Practical Hydraulics ; a Series Of Rules and T ablesfor the use of Engineers, etc ., etc . By THOMAS Box. Fifth edition,numerous p lates, post 8vo , clo th , 5s.The Essential EZoneenis of Practical M echanicsbased on the Principle of Warh, designed for Engineering Students. ByOLIVER BYRNE, formerly Professor of Mathematics, Co llege for C ivilEngineers. Th ird edition, with 148 wood engravings, post 8vo , cloth,7s. 6d.CONTENTSChap. 1 . How Work is Measured by a Unit, both with and without reference to'a Unitof T ime— Chap . 2 . T he Work of Liv ing A cuts , the Influence of Fric tion, and introducesone of the most beautiful Laws OfM otion hap . 3. The princ igles expounded in the first andsecond chapters are app lied to the Mo tion o f Bodies—Chap . 4. he Transmission ofWork bysimple Machines—Chap . 5. Useful Propositions and Rules.Breweries and M altings the ir Arrangement, Construction, Machinery, and Plant. By G . SCAMELL, F Secondedition, revised, enlarged, and partly rewritten. By F. COLYER,M I M E Wi th 20p lates, 8vo , cloth, 18s.A Practical Treatise on the Construction of H orizontal and Vertical Waterwheels , spec ially designed for the use of operative mechanics. By W ILLIAM CULLEN, M illwright andEngineer. l/Vzthn p lates. Second edition , revised and enlarged, small 4to , cloth , 1 2s. 6d.A Practical Treatise on M ill-gearing , Wheels, Shaf ts,Rzggers, etc. ; for the use of Engineers. By THOMAS Box. Thirdedition, with u p lates. Crown 8vo , cloth , 7s. 6d..Mining M achinery:a Descriptive T reatise on theMach inery, Too ls, and other Appliances used in M ining. By G. G .ANDRE, F Assoc . Inst. C .E. , Mem. of the Soc iety of Engineers.Royal 4to , uniform w ith the Author’s T reatise on Coal M ining, containing 182 p lates , accurately drawn to scale, with descriptive text,in2 vols., cloth, 31. 1 zs .CONTENTSach inery for Prospe c ting, Excavating, Hauling, and Hoisting— Ventilation—Pumpi'ngT reatment of M ineral Produc ts , including Go ld and Silver, Copper, T in, and Lead, IronCoal, Sulphur, China Clay , Brick Earth, etc .Tahles for Setting out Curves f or Railway s, Canals,Roads, etc., varying from a radius of five chains to three miles. By A.KENNEDY and R. W . HACKWOOD . I llustrated, 32mo , cloth, 2 s. 6d.PUBLISHED BY E. F. N. SPON.The Science andA r t of the M anufacture of PortlandCement, w ith observations on some of its constructive applications. Wi th66 il lustrations. By HENRY REID , C.E., Author of ‘A PracticalTreatise on Concrete,’etc ., etc. 8vo , cloth , r8s.The D raughtsman’s H andbooh of P lan and M apD rawing ; including instructions for the preparation of Engineering,Architec tural, and M echanical D rawings. W'i th numerous illustrationsin the text, and 33 p lates ( 15 pr inted in colours). By G . G. ANDRE,Assoc . Inst. C .E. 4to , cloth, 9s.CONTENTST he Drawing Offi ce and its Furnish ings—Geometrical Problems—Lines , Do ts, and the irCombinations— Co lours , Shading, Lette ring, Bordering, and N orth Po ints— Scales— Plo ttin—Civil Engineers’and Surveyors’Plans—Map D rawing— Mechan ical and ArchitecturaDra'sving—Copy ing andReducing T rigonometrical Formulas, etc . , e tc .TheBoiler -maher’s andIron Ship-huilder’s Companion ,comprising a series of original and carefully calculated tables, of theutmost utility to persons interested in the iron trades. By JAMES FODEN ,author of Mechanical Tables, ’etc . Second edition revised, with illustrations, crown 8vo, cloth, 5s.Roch Blasting :a Practical T reatise on the meansemployed in Blasting Rocks for Industrial Purposes. By G. G . ANDRE,Assoc. Inst. C .E. ”71th 56 illustrations and 1 2 p l ates, 8v0, c loth,tor. 6d.Painting and Painters’M anual:a Book of Factsfor Painters and those who Use or D eal in Paint Materials. By C . L.COND rT and J. SCHELLER. Illustrated, 8vo , cloth , ros . 6d.A Treatise on Ropemahing as practised in puhlic andprivate Rope-yards, with a D escription Of the Manufacture , Rules, TablesOf Weights, etc. , adapted to the Trade , Shipping, M ining, Railways,Builders, etc . By R. CHAPMAN, formerly foreman to Messrs. H uddartand Co . , Limehouse, and late Mas ter Ropemaker to K .M . Dockyard,D eptford. Second edition, 1 2mo, cloth, 3s.Laxton’s Builders’and Contractors’Tables for theuse of Engineers, Architects, Surveyors, Builders , Land Agents, andothers. Bricklayer, containing 2 2 tables, with nearly calculations.4to , cloth, 5s.Lur ton’s Builders’and Contractors’T ahles . Excavator, Earth , Land, Water, and Gas, containing 53 tables, with nearlycalculations. 4to , cloth, 5s .CATALOGUE OF SCIENT IFIC BOOKSSanitary Eng ineering:a Guide to the Constructiono f“forks of Sewerage and House D rainage , with Tables for fac ilitatingthe calculations of the Engineer. By BALDW IN LAT HAM , C .E. , M . Inst.C .E. , F Past-President of the Soc iety o fEngineers . Secondedition, with numerousplates and woodcuts , 8vo , c loth , 1 1. 10s.Screw Cutting Tablesf or Engineers and M achinists,giving the values of the different trains of Wheels required to produceScrews of any p itch , calculated by Lord Lindsay, M .P. ,etc . Cloth, oblong, 2 s .Screw Cutting Tables, for the use of MechanicalEngineers , showing the proper arrangement of Wheels for cutting theT hreads of Screws o f any required pitch, wi th a Table for making-theUniversal Gas-pipe Threads and Taps. By \V. A. MART IN, Engineer.Second edition, oblong, cloth, Is. , or sewed, 6d.A Treatise on a P ractical M ethod of D esigning SlideValve Gears by Simp le Geometrical Construction ,based upon the princ iplesenunc iated in Euclid’s Elements, and comprising the various forms ofPlain Slide -Valve and Expans ion Gearing ; together with Stephenson’s,Goo ch’s, and Allan’s Link-Motions, as applied either to reversing or tovariable expansion combinations. By EDWARD J. COW LING \VELCH ,Memb. Inst. Mechanical Engineers. Crown 8vo , cloth , 6s.Clean ing and Scouring :a Manual for Dyers, Laundresses, and for Domestic Use . By S. CHRISTOPHER. I8mo , sewed, 6d.A Glossary of Terms used in Coal M ining . ByW ILLIAM STUKELEY GRESLEY, Asso c. Mem. Inst. C .E., Membero f the North of England Institute ofM ining Engineers. I llustrated withnumerous woodcuts and diagrams, crown 8v0, clo th , 5s .A Pochet-Booh for Boiler M ahers and Steam Users ,comprising a variety of useful information for Employer andW orkman,Government Inspectors, Board of T rade Surveyors , Engineers in chargeo fWorks and Slips, Foremen of Manufactories, and the general Steamusing Public . By MAURICE JOHN SEXTON. Second edition, royal32mo, roan, gilt edges, 5s.Electroly sis a Prac tical T reatise on Nickeling,Coppering, Gilding , Silvering , the Refining ofMetals, and the treatmento f Ores by means o f Electricity . By H IPPOLYTE FONTAINE, translatedfrom the French by J. A. BERLY, C .E., Assoc . S .T .E. ifVith engravings.8vo, cloth, 9s.https://www.forgottenbooks.com/joinCATALOGUE OF SCIENT IFIC BOOKS.The Assay er’s M anual:an Abridged T reatise onthe D ocimastic Examination of Ores and Furnace and o the r Artific ialProducts. By BRUNO KERL. Translated by W . T . BRANNT . With 65illustrations, 8vo , cloth, I2s. 64.Electr icity :its T heory , Sources , and Applications .By J. T . SPRAGUE, Second edition, revised and enlarged, w ithnumerous illustrations, crown 8vo , c loth , 1 5s.The Practice of H ana’Turning in Wooa’, Ivory ,Shell,etc., with Instructions for Turning such Work in Metal as may be requiredin the Practice Of Turning in Wood, Ivory, etc . also an Appendix onOrnamental Turning. (A book for beginners.) By FRANC IS CAMPIN.Third edition, with wood engravings, crown 8vo , cloth, 6s.CONTENTSOn Lathes—Turning Tools—Turning Wood—D rilling— Screw Cutting—M iscellaneousApparatus and Processes—T urning Particular Forms— Staining—Pohshmg—Spinmng Metals—Materials—Ornamental Turning, etc .Treatise on Watchworh, Past ana’Present. By theRev . H . L. NEL’I‘HROPP, M .A., F.S.A. Wi th 32 illustrations, crown8vo , cloth, 6s. 6a'.CONTENTSD efinitions of Words and T erms used in Watchwork—T ools— T ime—H istorical Summary—On Calculations of the Numbers forWhee ls and Pinions ; the ir Proportional Sizes ,T rams, etc .—Oi D ial Wheels, or M o tion Work—Leng th of T ime of Go ing withoutWindingtip— T he Verge— T he Horizontal—The Duplex—The Lever—The Chronometer—Repeating\Vatches—Key less Watches- The Pendulum, or Sp iral Spring—Compensation— j ewe lling ofPiv ot Holes—C le rkenwell— Fallacies of the T rade— Incapacity ofWorkmen—How to Chooseand Use aWatch, e tc .Algehra Self- Taught. By W . P. H IGGS, M .A . ,D .Sc ., LL.D . , Assoc . Inst. C .E. , Author Of A Handbook of the D ifferential Calculus,’etc . Second edition, crown 8vo , cloth , 2s. 6d.CONTENT SSymbols and the Signs of Operation—The Equation and the Unknown QuantityPositive and Negative Quan tities— M ultiplication— Involution—Exponents— Negative Exponeuts—Roots, and the Use of Exponents as Logarithms— Logarithms— Tables of Logarithmsand Proportionate Parts— T ransformation of System of Logarithms— Common Uses ofCommon Logarithms— Compound Multiplication and the Binomial T heorem—Div ision,Frac tions, and Ratio—Continued Proportion—The Series and the Summation Of the SeriesLimit of Series—Square and Cube Roots—Equations—List of Formula . e tc .Spons’D i cti onary of Eng zneerzng , Ci vi l , M echan i cal,M'litary ,and Naval ; with technical terms in French , German, Italian,and Spanish , 3I00 pp ., and nearly 8000 engr avings, in super-royal 8vo,in 8 divisions, 51. 8s . Complete in 3 vols. , clo th , 51. 5s. Bound in asuperiormanner, half-morocco , top edge gilt, 3 vols , 61. 1 zs.PUBLISHED BY E. F. N. SPON.N otes in M echanical Eng ineering . Compiled princ ipally for the use of the Students attending the C lasses on this subject‘atthe City of London Co llege. By HENRY ADAMS, Mem. Inst. M .E. ,Mem. Inst. C .E., Mem. Soc. ofEngineers. Crown 8vo , clo th, 2 s. 6d.Canoe and Boat Building:a complete Manual forAmateurs, containing plain and comprehensive directions for the construc tion Of Canoes, Rowing and Sailing Boats, and Hunting C raft.By W . P. STEPHENS. W'ith numerous illustrations and 24 p lates qfWorking D rawings . Crown 8vO, cloth, 7s. 6d.Proceedings of theN ational Conference of Electricians,Philadelphia, October 8th to I3th, 1884. I8mo, cloth, 3s.Dynamo - Electricity , its Generation, Application,Transmission, Storage, and Measurement. By G. B. PRESCOTT. With545 illustrations. 8v0, cloth, I l. Is.D omestic Electr icityfor Amateurs . T ranslated fromthe French o f E. HOSPITALIER, Editor o f “ L’Electti c ien, by C . J.VVHARTON, Assoc . Soc. Tel. Eng . M unerous illustrations. Demy 8y o ,cloth, 9s.CONTENT S1 . Production of the Electric Current- a. Electric Bells—3. Automatic Alarms— 4. D omesticT elephones—5. Electric Clo cks—6 . Elec tric Lighters— 7 . Domestic Electric Li hting8 . D omestic Application o i the Electric Light- 9 . Elec tric Mo tors —To . Electrical comotion— xx. Electrotyp ing, Plating, and Gilding— i a. Elec tric Recreations—x3. Various appiications—Workshop of the Elec trician.Wrinhles in Electric Lighting . ByVINCENT STEPHEN.With illustrations. I8mo, cloth, 2s. 6d.CONTENTS1 . The Electric Current and its produc tion by Chemical means— 2 . Produc tion of Elec tricCurrents by M echanical means—3. Dynamo-Electric Machines— 4. Elec tric Lamps5 . Lead—6. Ship Lighting .The Practical Flax Spinner ; be ing a Description Ofthe Growth , Manipulation, and Spinning o f Flax and Tow. By LESLIEC . MARSHALL, ofBelfast. With illustrations. 8vo , cloth , I5s.Foundations and Foundation Walls f or all classes ofBuildings, Pile D riving, Building Stones and Bricks, Pier and W allconstruction, Mortars, Limes, Cements, Concretes, Stuccos, &c . 64 illustrations. By G. T . POWELL and F. BAUMAN. 8vo, cloth, ros. 6d.1 6 CATALOGUE OF SCIENTIFIC BOOKSM anualfor Gas Engineering Students. By D . LEE.ISmo , cloth Is.Hydraulic M achinery , Past and Present. A Lecturedelivered to the London and Suburban Railway Ofli c ials’Association.By H . ADAMS, Mem. Inst. C .E. Folding plate. 8vo , sewed, Is.Twenty Years with the Indicator . By THOMAS PRAY,Jun , C .E., M .E., Member of the American Soc i ety Of Civil Engineers2 vols., royal 8vo, cloth, 1zs. 6d.Annual Statistical Report of the Secretary to theM embers of the Iron and Steel Association on the Home andfibrezgu Ironand Steel Industries in 1884. 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SPON ,12 5 , Strand .N ew Y ork :12 , Cortlandt Street .SYNOPSIS OF CONTENTS.Indium.Iridium.Iron and Steel .LacquersandLacquering.Lanthanum.Lead.Lithium.Lubricants .Magnesium.Manganese.Mercury .M ica.Molybdenum.Nickel .Niobium.Osmium.Palladium.Platinum.Potassium.Rhodium.WORK SHOP RECEIPT S ,FOURTH SERIES,DEVOTEDMAINLYTOHANDICRAFTS MECHANICALSUBJECTS.BY C . G . 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N . SPON , 125 , Strand .N ew Y ork :12 , Cortlandt Street .mixed.Second, chemical explosives in which the ingredients areunited by a more or less unstable affinity.In the first case,the partic les which have to combine areseparated by apprec iable distances, whilst in the second,each molecule contains within itself the reagents, and is infact a complete explosive per se.The reaction in the second c lass is therefore much morerapid, and the violence of the explosive proportionatelygreat. Moreover, inmany of the second c lass, the chemicalunion is exceedingly unstable, andconsequentlymore subjectto the influence of external action .Of the first c lass, by far the most important is gunpowder,composed of nitrate of potassa,sulphur, and charcoal, theusual proportions in this country being 75, 10, and 15 respectively, and the same proportion, or very nearly so, has of lateyears been adopted by France andBelgiumfor all large guns.An exception must, however, be made with regard tococoa ”0powder, the composition of which is kept secret .It differs from the ordinary powders in the nature of thecharcoal, which is made from rye straw instead of the dogwood and alder generally used, and in the small proportionof sulphur.14. The composition and manufacture of explosives generally , is fore ign to my present purpose, as gunpowder is nowexclusively used for artillery purposes. Such powder is ofthe description commonly called nitrate powder, the oxidisingagent being nitrate of potassa.10 INTERNAL BALLISTI CS.There are , however, other descriptions of powder whichmay be briefly mentioned, viz15 . (1) Nitrate of Soda Powder.— If for nitrate of potassa,nitrate ofsodabe substituted in the same proportions, a powderis obtainedwhich is somewhat less costly and about one-thirdstronger than the nitrate of potassa powder.It is,however,more subject to absorbmoisture , although thisis probably due to the imperfect fabrication of the nitrate ofsoda, which when pure is not deliquescent, and probably onlyacquires that property by the presence of de liquescent salts,such as chlorides.Nitrate of soda powder was used to a very great extent inthe excavation of the Suez Canal.1 6 . (2) Nitrate of Baryta Powder. —This powder is of aslower combustion than nitrate of potassa powder, and cousequently strains the gun less severely . It has, however, theproperty of fouling the gun to a considerablv greater extent.17. (3) Chlorate of Potassa Powder.— W ith equal weightthis salt contains less oxygen than nitrate of potassa, but itdecomposes more readily andmore completely, consequentlythis powder is more violent. It is also somewhat dangerous,being liable to explode under a sudden shock, and is moreerosive to the iron and offensive to the servic e from thechlorine gas which is evolved.1 8. (4) Picrate of Potassa Powder.— Powder combinedof pic rate of potassa,saltpeti e , and charcoal was tried byM . Designolle in France in 1867 .This powder gave go « 1d results in torpedoes and also inguns,andhad the advantage of not evolving e ither sulphurousac id or sulphide of potassium,being so far of considerableadvantage in ships and casemates, but several cases of premature explosion having taken place , the use of the powderdoes not appear to have been persevered in . It is, moreover,costly, and the manipulation of picrate of potassa is somewhatdangerous.19 . (5) Picrate of Ammonia Powder . — This salt is lesssensible to a sudden shock than the picrate of potassa. Inhttps://www.forgottenbooks.com/joinI2 INTERNAL BALLISTICS.Cravimetr ic D ensity.23. This term has different meanings, in England and inFrance .24. In England, it is defined in the Text-Bookof Gunnery‘to be the ratio of the weight of a charge of powder in thechamber of a gun, to the we ight of that volume of waterwhich wouldfill the space behind the proj ectile.”Thus if gravimetric density 1 , the space occupied by1 lb. of the charge is the same as would be occupied by 1 lb.o f water,i. e . cubic inches, or in other words, if thecharges be spaced so as to allow cubic inches to thepound Of charges, the gravimetric density is unity.For any other spacing if n be the number of cubic inchesallowed to the pound of charge,Grav imetric denSityThe usual notation in England for denoting the co nditionof a charge is to write first the weight Of the charge , then thedesignating mark of the powder, and lastly the quotient ofby the number Of cubic inches allowed to the pound.27 73Thus 75 P30cubic inches per lb.25 . In France,this is called Densité de Chargement, asdistinct from Densité Gravimétrique,” by which latter termis meant,the weight in kilogrammes of 1 cubic litre of thepowder not pressed together, except by its own weight.Consequently Densité Gravimétrique 1,means 1 kilogramme occupying a space of 1 l itre or 1 dec imetre cube, orin our units 2 ° 2 lb . Of powder in a space Of cubicinches, or 1 1b. of powder In a space of 27 ° 73 cubic inches,which IS the same as i n our system.But it is to be noted, that one dec imetre cube of powderdoes not always weigh 1 kilogramme . The weight increasessignifies 75 lb . of P powder spaced at 30Text book of Gunnery,’by Major S. Mackinlay, p . 22,1887.INTERNAL BALLIS'TI08 . 13with the size of the grain, so that whilst with the smallgrained powder designated F2 in France , it is 934 to 944grammes with the large A%% it is 1150 grammes, and therelative densités gravimétriques are° 934 and 1 150.This distinction is not made in England, and our gravimetric density is equivalent, not to Densité Gravimétrique,”but to Densité de Chargement.Formand D imensions of Grain.26. As will be seen hereafter, the form and size Of grainhave very important effects upon the action of powder in agun .A few years ago the powder called E.L.G . was the onlypowder used for artillery purposes.Owing to the small Size Of the grain, the time Of combustion was exceedingly small,and the action violent. Atthe same time , owing to the small space between the grains,the ignition was slow and very irregular, and this, as will beshown presently, gave rise to abnormal pressure, called,faute dc mieua, wave pressure by our artillerists.By degrees, and by a sort of tentative process, the size ofthe grains was increased, though the form remainedmore orless irregular. Such were the P and P, powders.Subsequently moulded grains Of a regular formwere introduced, such as the spherical, cubical, cylindrical, and finallythe prismatic , and quite recently Mr. Quick has introducedthe form of cylindrical discs placed one on the top of theother,perforated with cylindrical holes and with radiatingor other channels on the face of the discs so as to facilitateignition.The influence of these different forms will be examinedhereafter.HydroscopicQuality.27. All gunpowder is more or less liable to absorbmoisture and thus to deteriorate in strength by long storage ,giving rise to corresponding irregularity in ballistic results,14 INTERNAL BALLISTICS.but even when first manufactured there is considerabledifl'erence in the amount of moisture in different powders .The following table gives the percentage of water invarious powders as determined by Noble and Abel inEngland, andMM. Sarrau and Roux in FranceEnglish1 17 1 61Frenchof water.I NTERNAL BALLISTICS. 15CHAPTER II.FIRED GUNPOWDER.Products of Combustion.28 . When a charge of powder is fired, about'43 per cent.in weight is converted into permanent gases, that is to say , intogases which when cooled down to ordinary temperaturesretain the ir gaseous condition.The remainder, or about 57 per cent. by we ight,is, whencooled down, a solid residue,but wh ilst in the gun at thehigh temperature of ignition, is in a liquid form diffusedthrough the gases. When the charge is fired in a closevessel and thegases are afterwards allowed to escape , th isliquid rapidly solidifies.29. The chemical composition of the products Of combustion has not much interest in a ballistic point of view,and the compounds, espec ially the solids, appear to varywithin very considerable limits. Those who are interested inthis question will find it very ably discussed by Messrs.Noble andAbel in their papers published in the Transactionsof the Royal Society in 1875 and 1879, and also in areport by MM. Morin and Berthelot in the Academic desSc iences Comptes Rendus,’The solids consist chiefly Of compounds of potassium, withcarbonic , sulphuric , sulphurous, and nitric acids.The gases are approximately represented as followsCarbonic acid 2596Carbon ic oxide 0343Nitrogen 1084Sulphurous acid 0099Marsh gas° 0003Hydrogen 0007Oxygen 0003Water 01480° 4283 per cent.16 INTERNAL BALLI8 TICS.30. By some it is believed that, at the very high temperature prevailing in a gun, the liquid residue is itself converted into the gaseous form, and that it gives out workduring its expansion, but it is maintained by Noble andAbel,that the work done by expansion is due only to the gaseousportion of the products, although these are sustained duringtheir expansion by heat imparted to them by contact withthe particles of liquid difi'used throughout the ir volume at avery high temperature. This hypothesis will be discussedfurther on .At present it may be added,that even if the 57 per cent.of solids be in a gaseous form,it has been shown bySch ischkofi'and Bunsen that the tension of such gases mustbe exceedingly small, and quite incapable of producing anyappreciable effect on the general tension Of the permanentgases.31 . It is shown by Noble and Abel that at the temperature of ignition,the volume of the 57 per cent. of inertliquid is approximately equal to that Of the powder fromwhich it is derived.D issociation .32. By some authorities it is maintained, that althoughthe various compounds determined by analysis are found inthe final state Of the produc ts, they do not exist in theearlier stages whilst the action is taking place in theprojectile .They say that these compounds cannot exist at the hightemperature there reigning, owing to the interference of thephenomenon called dissoc iation,”but can only arise whenthe temperature has so far fallen as to admit of suchsecondary combinations.33. Although there is no positive evidence of “ Dissociation in a gun, it may be we ll to examine what would bethe effect ballistically, if it did take place .INTERNAL BALLISTI03. 1734. It appears from. the table 29) that about 60 percent. of the gaseous products consist of carbonic ac id ; aswill be shown hereafter,the temperature of ignition is about2340°C. , and as the temperature Of dissoc iation of carbonicacid is about 1800° C.,it is held that the carbonic ac id couldnot exist until the temperature had fallen to 1800° C.,andthat up to this time only carbonic oxide could exist. Butthe question may well be asked, why should not the carbonicoxide be subject to dissoc iation as well as the carbonic ac id?If this were so , the gaseous products of combustion would beconfined to the oxygen liberated by the decomposition ofthe nitrate of potash, and Of course this would only takeplace at a lower temperature than 18000 C.Consequently , even if the whole of the carbon were converted into carbonic oxide, we would have the volume Ofthis gas and the volume of the remaining oxygen existingtogether so long as the temperature exceeded andthis would give a compara tive ly low pressure , until by thefall of temperature, the combination of the free oxygen withthe carbonic oxide would give rise to a sudden and verylarge increase of temperature , and a sudden almost explosiveinc rease of p ressure .The result would therefore be a low initial pressure ,dec reasing as the shot travelled towards the muzzle , andthen a very sudden rise of pressure, again falling till the shotleft the gun . This certainly does not represent the realconditions.35 . It is true'that the volume Of one equivalent Ofoxygen plus that of one equivalent of carbonic oxide isone and a-half times greater than that of the resultingequivalent of carbonic ac id,and th is would increase thepressure as long as dissoc iation took place , but the suddenincrease of pr. esure when the temperature fell to 1800°would equally take place .36 . But it may further he remarked, that it by no meansfollows, that because carbonic ac id may be dissoc iated at aten perature exceeding under atmospheric pressure,018 INTERNAL BALLISTICS.the same would take place under the enormous pressuresexisting in a gun .37. Another argument against dissoc iation may be thusstated.If a c in a gun, the length Ofwhich is representedbyAG,and the projectile be not allowedto move until the whole of the charge is burnt, the pressuremay be represented by a curve B C D which is Noble andAbel’s curve , and if the gun be suffic iently lengthened,thepressure would fall to zero at a point X on the same curve.If, however, the projectile be allowed to move, the pressurecurve will be an ascending curve Up to the point Of maximum pressure C when all the powder is burnt, after whichit will descend as before to X.The work done on the projectile will, therefore,berepresented by the area ACD G .Now if from dissoc iation or otherwise, the point of maximum pressure be removed further forward to E, the pressurecurve will be AEF X falling to zero at the same point X,and the work done on the projectile will be AEF G .Now the area AC D X must be equal toAEFX becausethe whole of the heat is expended in the two cases. ButFGX is greater than D GX, therefore A C D G must begreater than AEFG.https://www.forgottenbooks.com/join20 INTERNAL BALLISTI08 .42. Piobert further found, that if a train of powder,instead of being simply laid on a flat surface,is laid in agroove and covered with a light plank, the veloc ity oftransmission is about doubled when the groove is entirelyfilled with powder.If, however, it is only half filled, the veloc ity is stillfurther increased by about one-half.On firing trains in tubes he found analogous results, as isshown in the following tableWeigh t ofpowder per transmis of samelineal foot. siomn tube. train inlb. feetpersec . Open air.Formation of Train.Tube in diam.. 1 thickness of paper, full °0806416 full 08064 12 13716 half full °043680 785 16 full 21168 13 78043. From his very numerous and carefully conductedexperiments, Piobert arrived at the following conclusions(a) The veloc ity of Ignition varies very nearly in theinverse ratio of the fourth root of the diameter,or equivalent diameter, of the grain.(b) It decreases with the increase of density and thedegrees of glazing .(c) The composition (dosage) and length of triturati on ofthe powder do not appear to have any appreciable effect onthe velocity of Ignition .Combustion.44. By the Combustion of a grain is meant, the gradualburn ing downwards from the surface until the whole isconsumed.INTERNAL BALLISTICS. 2145. On this subject Piobert also made many eXperimentsand conc luded as follows(a) That cwteris paribus, the veloc ity of Combustion variesinversely as the density of the grain, or 1; 8 a, a constantwhich depends upon the nature of the powder, and denotesthe weight of powder burnt per second per un it of surface .(b) The veloc ity decreases rapidly with an increase ofhumidity in the powder.(a) It increases with the amount of trituration up to acertain limit.(d) W ith the same density, same trituration, and samehumidity, the greatest velocity of Combustion was obtainedfroma powder composed ofSaltpetreCarbonSulphur(e) In free air, the veloc ity of Combustion varied, inthedifferent powders used by Piobert, from 0°4 inch persecond to 0 6 inch per second.46 . When powder burns under pressure, as in a gun orclose vessel, the veloc ity of Combustion is very greatlyincreased. After careful examination of the results obtainedby himself and other artillerists, Mons. Sarrau adopts thefollowing formulawherevelocity of Combustion at pressure p .atmospheric pressure p o.That is to say, that the veloc ity varies as the square rootof the pressure .Thus, a powder which in the open air would burn with avelocity of inch per second, would burn with a veloc ityof about 33 inches per second under a pressure of 3000atmospheres.22 INTERNAL BALLISTICS.Ef eat of Rate of Ignition and Combustion on Pressure.47. Fromwhat precedes, it is obvious that the two phenomena of Ignition and Combustion have each adistinct partin the conversion of a charge of powder into its products,and that it is on their jointactions that the efi'ect offiring thecharge depends.48 . When a charge of powder, placed behind a projectile,is ignited at the rear end, the Ignition commences with thegrains at that end, and gradually extends throughout thecharge . If the rate of Combustion be great and the size ofthe grains small, it may be, that the" grains at the backare entire ly consumed before those in front are ignited, andthis is likely to take place with a small -grained powderc losely packed in a long tube , that is to say, in a long chargeof fine -grained powder.This rapid development of gas at the rear end of thecharge would give rise to a considerable local pressure andcompress the powder next the projectile,wedging it up as itwere into a mass of high gravimetric density, before the projec tile hadmoved to any considerable extent.On the other hand, if the cartridge did not quite fill thechamber, the Ignition would pass rapidly through the vacantspace , and the Combustion would take place almost simultaneously at both ends of the cartridge . The result of thiswould be the early displacement of the projectile and anincrease of space for the evolving gases, giving rise to a lowerpressure behind the projectile.The same effect would take place if a large-grainedpowder were used, as in this case the interstices betweenthe grains be ing large, the Ignition would pass rapidly, andthe early action of the pressure on the base of the projectilewouldmove it forward and increase the space .49. Rapidity of Ignition throughout the charge is therefore a matter of considerable importance as preventing localvariations of pressure .It may be promoted in various ways, viz. by increasingINTERNAL BALLISTI08 . 23the size of the grains, by perforations through the mass ofeach grain, as in the case of the cylindric and prismaticpowders,by leaving a space between the cartridge andthe top of the chamber,by commencing the ignition at thec entre of the charge, or by igniting it simultaneously atseveral places. Of course the more simultaneous the Ignition, the more rapid is the development of gas, and themore rapid the rise of pressure.50. The rate of Combustion has a greater effect as regardsthe pressure than the rate of Ignition . It is on it that therate of evolution mainly depends, but the form of the grainhas also a very important effect, and it is the rate of evolution of the gas which chiefly affects the pressure .51 . The rate of evolution is a function of the rate ofCombustion and of the surface under Ignition. We maytherefore say, that the pressure is a function of the rate ofCombustion and of the surface jointly. But the rate ofCombustion is itself a function of the pressure, so that in fact,the pressure is a function of the ignited surface, andmakingthe time the independent variable, the quantity of gasevolved and the pressure are functions of the form of grainon which the variation of the surface depends.52. There is,however, in the case of a gun a furthercomplication. The space is also variable,and this of courseaffects the pressure behind the projectile .53. With a very quick burning powder the projectile hasmoved very little at the time when the charge is entirelyconsumed, while with a slow evolution of gas, it has moveda considerable distance , thus greatly increasing the spaceand decreasing the maximum pressure .There is a contest going on between the rapidly increasingspace and the increasing evolution of gas. The formerincreases in an increasing ratio with the time, whilst thelatter, though it increases, does so in a decreasing ratio withthe time , because the ignited surface in every ordinary formof grain decreases as the burning goes on. Then again, theevolution of gas increases with the pressure, and is thus an24 INTERNAL BALLISTI08 .increasing ratio up to the point of maximum pressure, and adecreasing ratio as regards the time, after.54 . If, at the time of the maximum pressure, the wholecharge is consumed, the subsequent pressures behind the projectile will be representedby an adiabatic curve,and the workdone on the projectile easily ascertainable, but as regards thework prev iously done , it is obvious, fromwhat is said above,that the problem is an exceedingly complicated one .55. If the charge be not entirely consumed at the time ofthe maximum pressure , one of two things may take placee ither the maximumpressure may be sustained for a pe riod,owing to the effect ofthe increasing space being exactly compensated by the inc reasingevolution of gas (which, however, isnot possible with any ordinary form of grain), or the pressuremay go on falling, but less rapidly than the adiabatic curve,owing to the continued evolution of heat and gas by theremaining unconsumed powder.In this case it is evident, that at the time when the wholeof the charge is consumed, the pressure will be the same asin the case ofthe other powder and the subsequent pressuresfollow the same law. Up to this point the pressures mustalways be less, and therefore the whole work done on the projectile must be less with a slow than with a quick powder,andto obtain an equal ballistic effect larger charges must beused.This is confirmed in practice and it is also in conformitywith the thermodynamic law,that any thermal machinewhich works between given limits of temperature gives themaximum effect when all the heat is received at the highesttemperature and rejected at the lowest.”Formof Grain .56. The influence of the form of grain upon the pressureand evolution of gas is very great, and although the relationbetween them and the distance moved by the projectile isINTERNAL BALLISTI08 . 2 5very complicated, an approximate idea of the influence offormmay be obtained by considering the evolution of gas ina close vessel as a function of the time.57. Let it be assumed, that the composition, density, anddegree of humidity are the same , the only difference being inthe formof the grain. Let three forms of grain be considered, the spherical, cubical,and prismatic or cylindric ,with a central hole . Further, that the weight of the grainsis the same, and that the whole surface is simultaneouslyignited.58 . Since the space is constant, the pressure is a functionof the time , and may be considered as equal to someunknown power of it.The rate of burning is,as stated before, proportionalto some power of the pressure. We may therefore assumethe velocity of burning proportional to some function of thetime, orwhere n is the rate of burning in free air, t the time, on anunknown power of it.On this assumption the volume of powder consumedmaybe found as a function of the time, as followsSpherical Grain.59. Let d diameter of the grain ;s distance from the original surface burnt atthe time t ;S actual surface under ignition at t ;S = 1r (dd sThe veloc ity of burning at sd t’which by assumptionfl ( 1therefore( 1 to.26 INTERNAL BALLISTIOS.and integrating and observing that when t 0s 0,(mSubstituting thi s i n{ dz(mini)? ((m z? 2 (m 1) p.“and as the thickness burnt in d t is u (1 t” ) d t, making Vthe volume burnt,+ sr + 1 ( 1(m+ 1+ 2 (m t orIntegrating and observing that when t 0V 0, we getfinallytm'l' l_4 7rdnz m+ l2m+ 1 ( 2t + tis tan -f a fl antam-f l )3Cubteal Grain.60. In this case let at be the side of the cube,s as before ,then d 2 8 is the length of the side at t and 6 (dthe surface under ignition. This only differs from the caseof the spherical grain by substituting 6 for W, and makingd; the side of the cube instead of the diameter of thesphere .https://www.forgottenbooks.com/join28 INTERNAL BALLISTI08 .”2{2 7r (T 4 p)18 8{ R - 6 1r} .63. In the forms of grain above dealt with, the surface ofIgnition decreases as the Combustion proceeds. In the prismatic form,however, with the central hole,one portion ofthe surface,v iz. the ends and outside , decreases, while thecentral hole increases.64. There is, howe ver, another formpatented by Mr. Quick ,in which he has sought to give a greater proportion to theincreasing surface by making the powder into discs of thesame diameter as the chamber of the gun andpierc ing itwitha number of cylindrical holes, and these discs are so connectedtogether that when a number of them are made up into acartridge , the cylindric ' holes will be continuous throughoutthe whole length . The flat discs are moreover dished outon one of their surfaces, so as to make these surfaces alsosurfaces of Combustion.If then R radius of discp radius of the holesv number of the holes ;T the thickness of discthe increasing surface is2 v T ,the decreasing surface,2 7rRT 2 1r (R2v pz),and if a:be the distance burnt at the end of t, the increasing surface is2 7r v ( p a) (Tthe decreasing surface ,2 1r (R — x)2INTERNAL BALLISTI08 . 29or the total surface under ignition at t— 2 w) - (R — a) (T (R — az)2V (Pand as was shown before,tut -l» !a: n (t m 1we getB m 1mp ow- l )—2c(m 3 m+ 3(2m 3) t“ + 3 i3 'H' 32 m+ 3+3 iTz—r 3whereA m.B : 2 7r n2 { T (v 1 ) 4 (R0 = 2 m 3{ 3 ( 1 n} .65. The rate of burning has in all these cases been assumedto be a function of the time represented by u n (1where u is the rate of burning in free air, and the unit of timeis taken as the ten-thousandth of a second, therefore15 0004 in .To illustrate the formula I will assume three valuesofm.m cc giv ing v u or un iform rate .m g giv ingV: u ( l t*) increasing as t’.m 1 v u ( 1 +1 3) increasing as t.66 . The formula then becomesFor spherical and cubical grainwhen m - oc V = A tV At1°5B£2 t2°5 tsts t3 5 t4'5- 125‘30 INTERNAL BALLISTI08 .when m= 1,+ 0 + 1i fFor prismatic grain,when mwhen mt3 t3’ 5 £4 t4°5I 0§I1 H5I2 25I10-125;when mFor disc powder,when mh v . At”B‘3w en m (t I l -sI “2‘ I1 5II'76ts s $4 t‘"+ C (§I T-I5I2 25I10-125’when m V =— A (i +I Z I 2;It will be seen by comparing these expressions that theyonly differ in the coeffic ients, A, B, andC, and that the termsinvolving the power of t for any given value of m, areidentical, showing that the coefficients depend upon the formof grain .67. The following table gives the value of these coeffic ients,for the four forms of grain above considered.INTERNAL BALLISTI08 .Prismatic with one hole. Disc with v holes.4 1 11 3 ” Gn S’ -p) i -p)Sw n’ R 24 11’ s u’{2 1r (T - 4 p) 2 1 n’{ T (v - l ) - 4 (R - v p) }4 t u’ —S- 6 1r } 2 r u' i 3 ( l - r) }where 91 0004R radius of sphere,or disc , or in the case of prismatic , the radius of the inscribed c ircle ;p radius of perforating holes ;S side of cube or hexagon in prismatic powder ;T thickness of prismatic grain or disc ;v number of holes in disc .68. The relative evolution of gas, as influenced by the formof grain, may be shown graphically by diagrams constructedfrom the above formula.Let it be assumed, that in each of the three first forms ofgrain , the grain is of the same weight, and that each contains1 5 72 cubic inches of powder, corresponding to the followingdimensions, say,and let the disc powder be of the following dimensionsOuter radius of discRadius of holesThickness of discNumber of holesdiameterlength of side2 R across flats2 p diameter of holesS side of hexagonT thi cknessR outer radiusp radius of holesv number of holesT thickness32 INTERNAL BALLI8 TIUS.As the volume of this disc is fourteen times greaterthan that of one of the above grains, for the sake of comparison, the volume given by the formula must be divided bvfourteen, as has here been done .q . - ‘q - cup n otQ - ‘ Q - O - n - u- 0l\\olIcl sol069. Fig. 1 represents the volume of powder convertedinto gas as a function of the time, on the assumption ofm cc,or a uniform rate of burning.INTERNAL BALLISTI08 . 33FIG. 2.l 3A A[II I t),I iI u I ,I ./III,I 9 'Il’,[II I /I I I 11 I I II1, I- LFigs. 2 and 3 represent the same on the hypotheses, thatm i , andm 1 respectively.‘Fro. 3.Abscissae 1 inch mfioQo-Uths of a second.Ordinates 1 inch 1 cub. inch powder.The following tables show,in each case, the time of consuming 1 572 cubic inch of powder, in ten-thousandths of asecond.In these diagrams the curve 1 represents Prismatic:2 represents D isc ;3 represents Cubical Grain ; 4 represents Spherical Grain .34 INTERNAL BALLI8 TI08 .T ime of consumption of1 ° 572 cubic inch .Value ofm.at70. From this it appears, that whatever be the form ofgrain, the prismatic is that which burns quickest, and consequently, as far as mere form is concerned, it ought to givethe highest pressure .But practically this is not the case , partly because prismaticpowder is generally of a higher density,but chiefly becausethe ignition is slow at first, owing to the high glazing andthe small surface.The cartridges are so built up, that the grains fit intoeach other, so that practically the initial surface of ignitionis almost limited to that of the central holes. The first evolution of gas is therefore small, and the initial pressure risesslowly , and as soon as it is sufficient to overcome the frictionof the projectile and the resistance of the base ring, the projecti le moves away, and by the time the evolution of gasbecomes rapid, the projectile is already some distancealong the bore, and the increased space keeps down thepressure .71 . It will be seen from the diagrams,that the evolution ofgas ismuchmore uniformwith the disc and prismatic powdersthan with spherical and cubical . This is no doubt an advantage , but it is inconsistent with what is often asserted, viz .that with prismatic powder, the combustion continues a longway down the bore . this form is thatwhilst the evolution to thelimited surface of combustion, it inc reases very rapidly as thecombustion goes on, and thus, though the maximum pressurehttps://www.forgottenbooks.com/join36 INTERNAL BALLISTICS.one-half the distance between the holes, and calling this Awe must makeA T (v — 1)2 6 (v — 1)In the case of the disc powder above considered, A 08,v 19, R 3, p O' l , making use of which values wefind T 3 ‘ 49 inches, the required thickness.75. Thus, theoretically, it is always possible to regulatethe thickness of the disc so as to have an always increasingsurface of combustion, but probably this will not be possiblein practice, on account of the difficulty of obtaining uniformity of density in thick discs .It is,however, proposed by Mr. Quick to unite a numberof these discs by means of a very inflammable cement,and Ibelieve he hashad considerable success with this method.If this can be accomplished, I have no doubt that cart ridgesmade up of discs thus united, will give excellent results,combining a very rapid subsequent evolution of gas with acomparatively low maximum initial pressure, and suchcartridges made up with an envelope of fusible metal will befound very convenient, especially for quick-firing guns, thecartridge case disappearing with the products of combustionand requiring no extraction.P roducts of Combustion.76 . In 28) it was stated that about 43 per cent.of the weight of the products consist of permanent gases,and 57 per cent. of compounds which solidify at ordinarytemperatures. As the ballistic effect of gunpowder issimply a conversion of a portion of the heat evolved intomechanical force, by means of the expansive action of thepermanent gases, a gun is just as much a thermal machineas is a steam or an air engine.If then the initial andfinal temperatures of the gases couldINTERNAL BALLISTICS. 37be ascertained,the fall of temperature, subject to certaindeductions,would give the energy of the projectile. Thesedeductions are, the energy of the products of combustion, theenergy of recoil of the gun and carriage, the force requiredto give rotation in rifled guns, and the friction of the projectile and the escaping gases.77. This method of treatment of the ballistic problem isdue to Count St. Robert who devotes some space to it in hisTraité de Thermodynamique , ’Turin, 1870, chapter iii ., andan attempt to further develope it formed the subject of apaper presented by myself to the Institution of CivilEngineers in 1884,and published in the Minutes of Proceedings, vol. xxx.,part ii. It is further dealt with in asubsequent chapter of the present work .78 . The volume of permanent gases, and the units of heatevolved from the combustion of a given weight of powderare fundamental data in ballistic problems, and the followingtable gives these for several descriptions of powder, as determined by Messrs. Noble andAbel.79 . Table of cubic centimetres of gas and units of heatevolved per gramme of powderDescription of Powder.Cocoa— Brown prismaticSpanish pelletCurtis and Harvey No . 6WalthamAbbey , F .G.E.L.G .PebbleM ining powder80. It will be at once observed, that those powders whichevolve the most gas give out the least heat, which is no38 INTERNAL BALLISTI08 .doubt due to the disappearance of sensible heat, in giving thegaseous form to the products.As the pressure when confined in a close vessel is a functionof the temperature and of the volume jointly, the figuresgiven in the last column represent to a c ertain extent therelative pressures fromthe above powders. Thus theWalthamAbbey powders are pretty nearly equal, whilst the miningpowder, Spanish, and Curtis and Harvey No. 6 are somewhatless,and the cocoa the least of all. The Curtis’and HarveyNo. 6 and the mining powder are nearly the same , althoughthe proportions of gas and units of heat differ by nearly50per cent.81 . The proportions by weight of the gaseous and solid orliquid portions of the products are shown in the followingtable , also due to Messrs. Noble andAbel .Gaseous.Sol id orCocoa— Brown prismaticSpanish pelletCurtis andHarvey No. 6WalthamAbbey , F .G .E.L.G .Temperature of Combustion.83. If the spec ific heats of the various products of combustion were known, it would be easy to obtain the temperature of combustion, but unfortunate ly this specific heat,though ascertainable at ordinary temperatures, increases withthe temperature according to an unknown law. Consequently the temperatures as deduced from the spec ificheats at ordinary temperatures can only be taken as superiorlimits.INTERNAL BALLISTICS. 39They will, however, approximately represent the relativetemperatures of combustion, whichmay be foundas follows0 ) LetW be the weight of powder ;H t he units of heat evolved per unit of weight ;3 the mean specific heat of the productsT the temperature of the products ;= ws T, or TAnd ifw be taken 1 gramme84. The mean spec ific heat at constant volume, of theproducts of combustion at ordinary temperatures have beendetermined by Messrs. Noble and Abel , and the followingtable shows the corresponding temperatures, the units ofheat evolved be ing taken from the table85.Cocoa Brown prismaticMining powder86 . There can be little doubt that these are abovethe actual temperatures in a gun, but they probablyrepresent the relative temperatures. There is, however,another method of calculation which probably gives prettynearly the actual temperatures. It is obtained from aformula given hereafter,40 INTERNAL BALLISTICS.when Tois the absolute temperature taken from 273° belowzero of the Centigrade scale .f is the pressure of the gases arising from 1 kilog. ofpowder occupying at the temperature of combustion, 1 c entimetre cube of space .p atmospheric pressure or 1 033 kilog. per squarecentimetre.uo volume of gases from un it of weight.Now for pebble powder and E.L.G. the value off is about2615 kilog. per centimetre, and v0Therefore,273 X 26 15X 276or by the Centigrade thermometer 2504 273 2231°C .which is probably approximately true.87. If this be so the mean spec ific heatsmay be determinedby dividing the units of heat evolved by unit of weight, bythe temperature just found. This gives for pebble powder721 °42231O 322’which is about 75 per cent. higher than the spec ific heatmade use of in the tableIf the spec ific heats used in that table be increased in thesame proportion , the resulting temperatures will be2504°absolute ,Description of Powder. Temperature .Cocoa— Brown prismatic 2390° 0.Spanish pelletCurtis andHarvey No . 6WalthamAbbey, F .G .E.L .GpebbleMining powderThe following table is given by Captain Roa of theFrench Artillery, and is derived from the experiments ofMessrs. Noble andAbel, andMessrs. Roux and Sarrau.INTERNAL BALLISTICS. 41Gramme Volume ofunits of heat gas evolved, TemperatureDescription of Powder. evolved per cubic of combusgramme of centimetres tion.powder. per gramme .PebbleF .G .Spanish pelletCurtis and Harvey No 6Mining powderFine poudre de chasseOrdinary cannon powderMusket powder AMining powder88. The high temperature of combustion of cocoa powder,accompanied as it is by an increased weight of the solid orliquid residue, and also by the increased charges required tokeep up the ballistic effect, may probably have a good dealto do with the rapid erosion of the bore in modern artillerypractice .Strength of Powder .89 . By the term “strength is denoted the pressure perunit of surface which the products of combustion exert onthe sides of a close vessel which is filled with the powder atgravimetric density 1 or inches per pound ofpowder .The absolute strength under these c ircumstances has notyet been ascertained, because in all experiments made tothat end,a certain amount of the heat evolved passes intothe substance of the containing vessel, and consequently islost as regards ballistic power. The proportion thus lostcannot be a constant ratio, because whilst the heat actuallyevolved is directly as the weight, or as the cube of the linealdimensions of the vessel, the surface exposed varies as thelineal dimensions, as regards the sides, and as the square ofthe lineal dimensions as regards the ends. Consequently,the loss of heat is greater in proportion in a small vesselthan in a large one .42 INTERNAL BALLI8 TICS.90. Much uncertainty exists with regard to the ac tualamount of cooling in a gun due to the transmission of heatto the metal .Count St. Robert, by his experiments on small arms, conc luded that about one- third of the whole heat evolved wasthus absorbed, or about 250units per kilo. of powder.MessrsNoble and Abel who experimented with a 12pounder, estimated the loss at about 100 units,and in a10- inch gun at not more than 25 units per kilog . of powder,being about 14 per cent. and 3s per cent. respectively .91 . On this subject M. Sarrau has made some interestingremarks which may be stated as follows.92. Let w be the weight of powder burnt in a closevessel .Tothe initial temperature of combustion .T the temperature of the products at any time t.a“ the surface of the vessel.v the rate of flow of heat in units per unit ofsurface and unit of time.Un ities:dec imetre,kilogramme, second, French unit ofheat, degree Centigrade.93. The combustion is not instantaneous but progressive,and at the end of any time t the quantity burnt will besome function of t. Let this be denoted by F (t).At the time t the heat absorbedmay be represented byOn the other hand, the heat lost by the products in fallingfrom To to T1 isc F (t) (T, T),a being the specific heat under constant volume, thereforec F (t) (To T)https://www.forgottenbooks.com/join44 INTERNAL BALLISTICS.and developing z in the same form, or makingand substituting in (6) the coeffic ients may be determined.97. At present,we will suppose that the law of burning isuniform, then if 7 be the time of total combustion of a grainand of the chargemaking(8) becomeswhich is satisfied by attributing to z a value independent oft such that x a"w -z,798 . The hypothesis of a uniform rate of combustion is aswe know incorrec t, because in a c lose vessel the pressureincreases as the time, and the rate of burning inc reases asthe pressure ; on the other hand, the surface of the grainsgenerally decreases as the time increases, so that there maypossibly be no great error in the hypothesis of uniformity .99. The equation (11) is easilv solved when It is known,for taking the logarithmslog s + z log a log x + log-r log u ,H e e'TcG O Tlog z + z log a log (INTERNAL BALLISTICS. 45log x log H + log e log e'+ log