F. A. P. Barnard on the Explosive Force of Gunpowder, 245 
by the cartridge, and the weight of the projectile. All these 
data, except those which relate to the gun and projectile, are 
furnished by the as just cited; and the remainder 
ay be deduced or directly taken from any table showing the 
initial vege obtained by experiment, and the gerne rs. of 
eWithou s by means of which rtd were obtaine 
of the mixed gases during expansion depen on the ratio of 
their capacities for heat at constant pressure and ret ye 
volume: and of this nothing had been oeevisiesly kno The 
formula of Hutton and the formulz in present use, for ealsuiag 
ing the initial velocities of cannon balls, are a Be law 
of Mariotte for the relation of the pressure of a us body 
to its density, This law furnishes a curve of ceaeating 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 compen- 
Sate 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 selection from these examples. 
In order to obtain a formula suitable for the purpose, we s ap. 
~~ a to represent the length of the space, measured along t 
re, which the liberated gases fill, provided they are steals 
Set free before the shot begins to move; 2, the variable length, 
measured in like manner, Ai they fill at any time after 
motion has commenced; "FE the initial foree by which the ahos 
is urged; v, the velocity acquired, and 7 the ratio between the 
€apacities for heat of the gases as taken at constant 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 cireumlocution. The term thermo-dynaimic index ap- 
pears to be sufficiently significant, and is believed not to be pre- 
Cccupied. It is therefore employe in the following discussion 
to denote the ratio in ques The conditions of the roblem 
give us immediatel y the ee which is founded on Poisson’s 
well known law for the pressure of expanding gases : 
dv=F(S)’ dt. And we have also de—=vdt, 
Hence ndo=F (2 )’ dz. And v?= oe rat aba40. 
os 2Fa 
1 aaa 
But when z==a, v0; and Ca—+ 
