999 
MESSRS. A. E. H. LOVE AND F. B. PIDDUCK 
p'/p is the ratio of the momenta of gun and projectile, that is (M+-|C)/M, so that the 
pressure ratio is approximately ^ = O'893. The agreement is to be expected ; 
for Table I. shows how little, relatively, yjy 0 varies with y 0 , so that Lagrange’s approxi¬ 
mation leads to little error in the total energy and momentum. 
44. Calculation of Recoil. —Prof. Love’s theory also enables us to calculate the distance 
recoiled by a very heavy gun while the projectile is travelling to the muzzle : this is 
important since the distance can also be found experimentally. We take from Plate 2 
the values of p' and p at intervals of 0 • 0005 sec. to the muzzle, and calculate j p' dt 
and p dt by approximate integration. These quantities are proportional to M/W 
and MV, where M', M are the masses of gun and projectile and V', V their velocities. 
A second integration gives M'S' and MS, where S' and S are the distances travelled by 
gun and projectile. For the muzzle epoch we find, in the present problem, M'S'/MS = 
1 /0-879. The recoil distance S' of the gun is therefore the same as for a massless pro¬ 
pellant and a projectile of mass M/0-879 = 56-9 kg., an addition of 0-57 times the 
mass of the propellant to that of the projectile. Lagrange’s approximation gives 
0 -5. Cranz* measured the recoil distance of a rifle, with comparatively slow combustion 
of the propellant, and obtained factors 0-496, 0-497, 0-477, mean 0-493. The theory 
of limiting motion would seem to apply with almost equal force to the case of slow 
combustion ; and thus we may regard Cranz’s experiment as confirming the recoil 
factor | and therefore (indirectly) the energy factor 1/3. Prof. Love has worked out 
the energy factor for a light projectile of mass 25 kg., and 12 kg. propellant, at epochs 
corresponding to (4) and (8) in Table I. The values are 0-335 and 0-333. 
45. A Special Solution of the Hydrodynamical Equations. —Prof. Love’s theory having 
suggested the possibility of the motion tending to a limiting form, it remains to show 
that the hydrodynamical equations admit of a particular solution in which the pressure 
is of the form f(y 0 ) <j> (t). We shall see that the pressure ratio and energy factor corre¬ 
sponding to this exact solution agree closely with those already calculated, and thus 
support is lent to the view that the limiting motion would be developed sooner or later 
with other initial conditions, e.g., with gradual introduction of gas from a burning 
propellant. If y t) is, as above, the initial distance of a particle from the breech and y 
its distance at time t, the general hydrodynamical equation is 
1 
^2 
C y , ^ 
po Vs = po [ -I 
ct \p 0 
r 
(i 
oy 
po cyo 
-Y-l l O 
° y 
po 
Write temporarily x = y w z = y—p 0 y Then 
02 
c z 
dt“ 
YP o 
po 
(1—po)" 
o2„ /0„ 
0 z cz 
dx \ dx. 
-Y-l 
* C. Cbanz, ‘ Zeitschr. f. d. ges. Schiess-u. Sprengstoffwesen,’ vol. 2, p. 345, 1907. 
