610 TRANSACTIONS OF THE AMERICAN INSTITUTE. 



maximum pressure. This assumption is, that the powder is completely 

 fired before the |)rojectile begins to move. The same assumption was made 

 by Robins and by Ilutton, and it is implicitly involved in all the velocity 

 formula which are found in treatises on artillery or on ballistics, at the 

 present time. It being assumed, then, that all the gas which the powder 

 is capable of producing is set free before the ball begins to move, we re- 

 quire to know, in order to determine the velocity it will generate in the 

 projectile, the following particulars, viz: the original bulk at the atmos- 

 pheric temperature and pressure, its capacity for heat both at constant 

 pressure and at constant volume, the length of the bore of the gun, the 

 part of this length occupied by the cartridge, and the weight of the projec- 

 tile. All these data except those which relate to the gun and projectile, 

 are furnished by the investigation just cited; and the remainder maybe 

 deduced or directly taken from any table showing the initial velocities ob- 

 tained b}^ experiment, and the dimensions of the gun by means of which 

 they were obtained. 



Without such an investigation no determination of the probable maxi- 

 mum pressure of gunpowder wdiich could be deduced from the observed 

 velocities of projectiles thrown by it, could be entitled to any confidence. 

 The rate of diminishing pressure of the mixed gases during expansion, de- 

 pends on the ratio of their capacities for heat at constant pressure and at 

 constant volume, and of this nothing had been previously known. The 

 formula of Hutton and the formula in present use for calculating the inia- 

 tial velocities of cannon balls, are founded on the law of Mariotte for the 

 relation of the pressure of a gaseous body to its density. This law fur- 

 nishes a curve of pressures in which the ordinates diminish as the bulk 

 increases much less rapidly than the real pressures; and accordingly, for 

 the production of a given effect, it makes the higher pressures too low to 

 compensate for the excess of the lower. 



The United States Ordnance Manual furnishes a variety of examples of 

 the initial velocities observed in firing round shot from smooth bore guns 

 of different calibres. The calculations which follow are founded on a selec- 

 tion from these examples. In order to obtain a formula suitable for the 

 purpose, we suppose a to represent the length of the space measured along 

 the bore, which the liberated gases fill, provided they are entirely set free 

 before the shot begins to move; x the variable length, measured in like 

 manner, which they fill at any time after the motion has commenced; F, 

 the initial force by which the shot is urged; v, the velocity acquired, and 

 g, the ratio between the capacities for heat of the gases as taken at con- 

 stant pressure and at constant volume. This ratio requires to be so often 

 referred to that it seems to be desirable to have some mode of indicating 

 it without circumlocution. Tlie term ther mo- dynamic index, appears to be 

 suflSciently significant, and is believed not to be preoccupied. It is there- 

 fore employed in the following discussion to denote the ratio in question. 

 The conditions of the problem give immediately the following, which is 

 founded on Poisson's well known law for the pressure of expanding gases: 



dv=^F( ~ydt. And we have also dx=vdt. 



