ON LAGRANGE’S BALLISTIC PROBLEM. 
225 
where 6 is given by the equation already written down. Using the reduction 
formulae we find 
e — m sin 2 A 
(m + 1) e sin 2 # ’ 
giving a = 0'325 when m — 11 and e = 12/50. The expansion formula, to the first 
power in e, is 
“ — rib e - 
46. Application to Ballistics .—To resume, Prof. Love’s theory supports the factors 
-o- and | up to considerable values of C/M, and shows further that the ratio of the pressures 
on projectile and breech (Plate 2) begins at once to oscillate about its mean value, 
reaching its first minimum when the projectile has travelled a distance of only two-thirds 
of a calibre. We may remark that no support is lent to the theory which appears to be 
favoured by Charbonnier* of more or less violent impulses of pressure on the base 
of the projectile : the discontinuity is at most one of pressure gradient, which becomes 
less and less as the motion proceeds. What would happen with gradual introduction 
of gas from a burning propellant is more conjectural, but nevertheless it seems of interest 
to examine the consequences of the assumption that the limiting state of motion, contem¬ 
plated above, is developed almost at once, and maintained ever after. The considerations 
which we shall advance have no pretence to rigour.f 
It is usual to measure maximum pressures in guns by crusher gauges placed at or near 
the breech. Let P be the pressure at the breech, P(l— C/2 M) that at the base of the 
projectile at any time, powers of C/M above the first being neglected. Compare the 
actual motion with that for a massless gas of the same thermodynamical properties, 
and a projectile of mass m. Then for identical motion of the two projectiles, with 
m/ M = 1+C/3M, 
p _ m 
P(l—C/2 M) “ M’ 
or p/P = 1 —C/6 M. In order to keep up the parallelism of motion we have to ensure 
that equal quantities of propellant are burnt in equal times. The rate of regression of 
the surface of colloidal propellants at different pressures has been measured by Vieille 
in a famous research.^ Mansell, who examined cordite M.D. by Vieille’s method,§ 
found a rate of regression in a closed vessel approximately proportional to the pressure. 
If D and d are the diameters of cordite in the two cases (or more generally numbers 
proportional to the linear dimensions of the grain), equal generation of gas corresponds 
approximately to the condition 
d _ p 
D ~ P (1 — C/4M) ’ 
* P. Charbonnier, ‘ Traite de Balistique Interieure,’ Paris, 0. Doin, p. 91. 
| See also F. Gossot and R. Liouville, loc. cit., pp. 50-58 ; vol. 17, pp. 61-66, 1914. 
J P. Vieille, ‘ Memorial des Poudres et Salpetres,’ vol. 6, p. 256, 1893. 
§ J. H. Mansell, ‘ Phil. Trans.,’ A, vol. 207, p. 243, 1908. 
