VOIy. 6, 1920 
PHYSICS: A. G. WEBSTER 
655 
If we now add (9) multiplied by (8) to (12) we have 
dQ = {ni + \w)udu + d{wzU) + - pdz + Spodx (13) 
b 
and if there are H units of heat evolved for each unit of mass of powder 
burned, and dq lost to the gun by conduction, we obtain the equation of 
conservation of energy, 
Hoidz — dq = {rn + \ic)udu -\- d{(j3zU) + - pdz -f Spodx (14) 
8 
Now if the specific heats are constant, we have 
t/ = c-„r = = - (15) 
K — I K — 1 
where k is the ratio of the specific heats. The quantity of heat Hdz — 
pdz/ 8 may be determined by calorimetric experiments in a bomb at con- 
stant volume, as is seen by putting dx and du equal to zero. Putting this 
equal to fdz/{ k— 1) as is customary, neglecting q and po, which can be 
done, and integrating (14) we have 
- {ni H- Xco) ^ + (16) 
K - 1 2 ^ _ 1 
which is Resal's equation, given in 1864. Combining this with (6) we 
eliminate the unknown temperature, which has never been experiment- 
ally determined, and obtain 
j(X)Z — - (/c — l)(w + 'Ko})u^. 
■ f= ' 2' (17) 
c' -\- Sx — co(l — z)/8 — corjz 
From this, if p, x, and u are experimentally determined, we may calculate 
z for each position of the shot. 
If we could neglect m"^, or if it were proportional to z and if x, p, z, were 
taken as coordinates, this would be the equation of an hyperbolic para- 
boloid, of which the sections z = const, are called isopyric lines by Lieut. 
Col. Hadcock in his paper on Internal Ballistics {Proceedings Royal Society, 
July 1, 1918 (479-509)). Col. Hadcock states that the expansion is 
neither adiabatic nor isothermal, but something between the two. This 
is true enough, as will be shown later, but it can hardly be justifiable to 
put, with any value of e. Col. Hadcock' s equation (5) p{v — otf jz^ = K, 
for this makes the pressure proportional to z\ instead of a linear fractional 
function of z, The assumption that u^/z is constant is not exactly true, 
but might be adopted without great error, as we shall see. 
