656 PHYSICS: A. G. WEBSTER Proc. N. A. S. 
If we now put for brevity, 
- - - = 5, - = C, - - = AZ_^ = E, (18) 
CO 5 ^ b m -\- \oi 
Ro + Sp, 
where B and C are constants depending on the load, D a constant of 
the powder, E a constant of the gun and shot, and G a mixed mass and 
specific heat constant, or g = f — Gu^jz an approximate constant, we 
may write 
jz - Gx" 
B Cx - Dz 
(19) 
p = (20) 
B + Cx - Dz 
We have heretofore said nothing about the rate of burning of the powder. 
After the powder is all consumed, z = 1, and the expansion is adiabatic. 
We are interested in the period of combustion. It is now customary to 
assume, after Charbonnier, that the rapidity of burning of the powder 
is a function of the amount already burned, the function (p{z) being known 
as the form function of combustion, while a factor A is called the vivacity, 
being, for a given powder, inversely proportional to the linear dimensions 
of the grain of given form. The rate of burning is also proportional to 
some increasing function of the presssure, let us say P{p). The simplest 
functions, except linear functions, are some powers of the variables, not 
in this case integral powers. If the power is progressive we take (p{z) = 
z^, if degressive (as in our powders), (p{z) = (1 — z)^. We will also put 
P = p", where a is positive, as the burning is faster the greater the 
pressure. We accordingly put 
y = Av{z)P{p) = ^(1 - (21) 
at 
We thus have the three simultaneous equations, 
dt = = ^ = (22) 
A<p{z)P(p) u E{p - F) 
It is obvious that the simplest choice for independent variable is z, accord- 
ingly we take 
doc ^ u ^23) ^ = ^ (24) 
dz A<p{z)Pipy dz A<p{z)Pipy 
and eliminating u by differentiation, 
