21G 
MESSRS. A. E. H. LOVE AND F. B. PIDDUCK 
we may assume that the required point (r, s ) is on the straight line joining the two 
extreme points whose co-ordinates have been determined previously. 
Alter these preliminaries the way is prepared for the numerical computation of any 
special case. 
PART II. 
41. Numerical Constants. — Prof. Love’s investigation was undertaken in order to 
throw light on a vexed question of internal ballistics, namely, how the mass of the 
propellant should be taken into account in calculating the velocity and pressure in a 
gun. Its completion has been delayed not only by the analytical complexity of the 
problem, but also by the time required for the numerical computations. In his original 
paper Lagrange set out from a certain state of the gas assumed as a first approximation, 
namely, one in which the velocity, at a given epoch, changed uniformly from one end 
of the gas to the other. Restricting attention to the case of a very heavy gun, the 
total momentum of gas and projectile is then (M + |C)V and the total kinetic energy 
L(M+^C)Y 2 , where V is the velocity of the projectile, M its mass, and C that of the 
propelling charge. Lagrange recognized that this state of motion is dynamically 
possible only in the limiting case of small charges, but made no real progress towards 
the theory for finite charges, the development of analysis being then inadequate to the 
problem. Since the ratio C/M in modern guns, though less than with gunpowder, is 
still of the order the importance of a full numerical discussion of Lagrange’s problem 
is evident. The calculations which follow were begun by Prof. Love, who determined 
all the fixed coefficients and the position and velocity of the projectile at various epochs. 
After verifying these figures I undertook the calculation of the distribution of pressure 
in the gas, at the times when a new type of wave was either being generated or extin¬ 
guished, and at the half intervals. Instantaneous combusion is assumed, as it appears 
hopeless to attempt to allow for the gradual burning of the propellant which occurs 
in actual guns. 
It is assumed that the propellant is cordite M.D., for which the maximum pressure 
for different densities of the gas, after explosion in a closed vessel, has been measured 
by Noble.* The results at medium pressure are represented approximately by the 
formula 
Pol 1 -l) = 9500, 
\po 1 
giving the pressure p 0 in .kilograms per square centimetre when p 0 is in absolute measure. 
This is the formula used in calculating initial pressures. The subsequent expansion of 
the gas is adiabatic, and will be represented by an equation of the form 
/1 \r 
_£> (-1 ) — const. 
V / 
* Sir A. Noble, ‘ Phil. Trans.,’ A, vol. 205, p. 201, 1906. 
