ON LAGRANGE’S BALLISTIC PROBLEM. 
177 
by the arc AC and the lines CP and PA. 
integral is 
I V 
J CP 
cZ nZ ' 
\ ds + r + s 
ds, 
which may be written 
[vzi- [vz] c - f 
JCP 
OS 
r + sj 
The contribution of CP to the line- 
~.c 
The contribution of PA to the line-integral is 
L 
av nV 
dr 
r + s 
dr. 
- Y 
Fig. 1. 
Now we can find a function to satisfy the equation for V, to make V = 1 at P, and 
so that, along CP, where r has the same value as at P, 3 V/ds = nV/(r + s), and along 
PA, where 6- has the same value as at P, 3V/3r — nV/(r + s). Then the value of 
Z at P is 
[VZ] C - 
AC 
V 
az ■ "Z-U+zf---^ 
\ or r + sj 
o + , 
cs r + sj 
) dr. 
The required function V can be shown, after Riemann, to be given by the equation 
V = , vB) *<«> !-«- l.f), 
/ | o 
where F is the symbol for the hypergeometric series, 
(r-F) js-s') 
{r + s ) (r' + s') ’ 
and r', s' are the co-ordinates of P. 
It may be observed that if n is an integer the series terminates, and V like Z is 
expressible in a finite form. 
It will be useful hereafter to note the formulse 
0V nV 
dr i ■ + s 
3V nV 
ds r + s 
_ (r + s) 11 ~ (s + /) (,s- — s') y / 
(r' + .s') n+1 dr 1 
_ (r + 6-)'‘~ 2 (r + s') (r — r') d y , 
(r' + s') n+1 dr K ’ 
1 —n, 1, £), 
1 —n, 1, 0- 
The Progressive Waves in Lagrange’s Problem. 
10. The Progressive Wave from the Left .—Let the positive sense of the axis of x be 
from left to right, and let the initial positions of the two pistons be given by x 0 = 0 
and x 0 — c, where c is positive. We shall denote the mass of the piston at x 0 = 0 by 
M, and that of the other piston by m. 
2 c 2 
