ON LAGRANGE’S BALLISTIC PROBLEM. 
217 
It appears probable, for various reasons, that the mean adiabatic index y is in the 
neighbourhood of 1-2. As we are restricted to a special set of values the value 
11/9 = 1-22 is selected. 
The problem discussed in detail is that of a gun of 15 cm. calibre, mass of projectile 
50 kg., charge of propellant 12 kg., distance travelled by the projectile from its initial 
position of rest to the muzzle 6 metres, initial volume of gas behind the projectile 
(chamber capacity) 30 litres. It is not, of course, possible with instantaneous combus¬ 
tion to keep the maximum pressure the same as it would be in a gun, though the muzzle 
velocity is much the same. The maximum pressure in this case is 6333 kg./cm. 2 . Had 
the pressure been kept down to 3000 kg./cm . 2 by taking a smaller charge, the problem 
would have been less representative as regards muzzle velocity, and as regards the ratio 
of the masses of propellant and projectile. 
In order to exhibit both the pressure in the gun and the degree in which the back 
particles partake of the motion of the projectile, eleven planes are taken at equal distances 
apart in the undisturbed gas, the end planes coinciding with the breech and the base 
of the projectile respectively. The horizontal line at the top of Plate 1 shows their 
initial positions. These eleven planes of particles are traced throughout their motion. 
The particles originally half-way between the breech and the base of the projectile may 
be called the middle particles,* and we shall choose, as epochs for the curves of pressure 
(Plate 1 ), the times at which a “ junction ” is either at the breech or the base of the 
projectile, or has just reached the middle particles. A junction is marked with a black 
circle on the figure. 
42. Details of the Calculation (Plate 1 , curve 1 ) (Article 10 ).—The early stages of 
the calculation call for no comment. We have <x 0 = 960,536-7 cm./sec., p 0 = 0-4, 
p 0 = 9500yo 0 /(l —/j 0 ) = 6333-3 kg. /cm. 2 , c = 339'5305 cm. (the initial distance from 
the breech to the base of the projectile is \c = 169'76525 cm.), a — 177,877-1 cm./sec., 
h = 778-0909 cm. The progressive wave which starts out from the base of the projec¬ 
tile reaches the middle particles (x 0 = \c) at time t = 0-0004772 sec. Particles between 
there and the breech are still at rest : from these particles to the base of the projectile 
the velocity of the gas increases almost uniformly to the value 99-6 m./sec,, and the 
pressure falls to 5651-3 kg./cm. 2 . The fall of pressure is remarkable considering that 
the projectile has only moved a distance of 2-4 cm. from its initial position ; and we 
observe a finite discontinuity in the pressure gradient on the two sides of the junction. 
(Plate 1 , curve 2 ).—The progressive wave reaches the breech at time t = 0-0009544 
sec., when the projectile has moved a distance of 9-28 cm. from its seat and has a 
velocity of 187-7 m./sec. The pressure falls from 6333-3 kg./cm . 2 at the breech to 
5097-2 kg./cm . 2 at the base of the projectile. 
(Plate 1 , curve 3 ) (Articles 12 , 16-17).—The first middle wave begins at the epoch 
just mentioned, by reflexion of the progressive wave at the breech. To find when it 
* These particles must be distinguished from those of the “ middle section of the theory, which here 
correspond to the breech of the gun. 
2 H 2 
