Vol. 6, 1920 
PHYSICS: A. G. WEBSTER 
657 
d^x _ 1 du 
dz^ ~ AipP dz 
dxy 
E(p - F) 
A'^cp'^P'' 
dx/P^dp ^ 
dz\P dz 
-)■ 
Now by logarithmic differentiation of (19) we obtain 
Idp^ 
p dz 
or by means of (23), (24) 
dz 
fz — Gu^ 
dz 
D 
(25) 
(26) 
B 
Cx — Dz 
dp 
dz 
f - 2GE(p - F) 
dx 
dz 
dz 
D 
fz - GA^<p^P 
B + Gx 
Dz 
(27) 
so that we have finally 
dH 
dz^ 
Ejp - F) 
AVP^ 
dx 
dz 
f - 2GE{p - E) 
dx 
dz 
dz 
D 
(dxY 
B Gx - Dz 
(28) 
This is a sufficiently complicated equation of the second order and of 
the third degree in dx/dz, but it is exact, and no simplifying assumptions 
have been made. If we know (p, P, p being obtained from equation 
(19), it gives x, the travel of the shot, in terms of z, the fraction of the 
powder burned. The equation (28) may be integrated graphically. When 
X is known as a function of z, p may be so determined, and then t and m 
from equations (22). Thus x, p, u, and z may all be found in terms of 
t, and the direct problem is solved. The equation (28) will be consider- 
ably simplified if we neglect G, w^hich is equivalent to assuming the 
temperature constant, which is usually done. 
We shall however use the differential equation (28) or rather (25) to 
solve the inverse problem. If x, p, and u are experimentally given, from 
which z is calculated by (19), everything in (28) is known except P and (p. 
If we begin with an approximate value of (p, say (1 — z)^ and an assumed 
value of p", equation (28) will be a linear equation for a and /3, from which 
by a few trials, the exact values may be obtained. 
We now come to the experimental portion of the work. The pressures 
in the rifle were observed by means of the gauge described in these Pro- 
CEKDiNGS for July, 1919, the film being afterwards placed in the lantern 
and an enlarged tracing being made with a pencil on squared paper. From 
