ON LAGRANGE’S BALLISTIC PROBLEM. 
209 
Also we have 
;-z 3 = f 
<r ’ s) dZ 7 dZ , 
5 — du + — dcr, 
(R,, S ,)OU GO~ 
where the integral is taken along the locus, and, since dZ/dcr vanishes along the locus, 
we get 
f(r,.) f* rlf 
= ut — U 2 T 2 — tdu — u —ds , 
“ V,s 2 ) Js. 2 ds 
ry* __ rp 
iA >2 
in which 
u = U 2 + 2 2 {(B' 1 -1)^+By 2 +By 3 + ... }, 
t = T 2 + Ci^ + c 2 S 2 + cf? + ..., 
5 = S 2 +t 2 S. 
It follows that, at x 0 = 0, x is given by the equation 
x X 2 = UaCi^T {U 2 c 2 +i - — 2 C 1 (Bfi— l)} c > 2 + {U 2 C 3 + fp 2 c 2 (Bfi — l) +-§fS 2 CiB / 2 } S" 
+ {U 2 c 4 + f£ 2 c 3 (Bh —l) + p 2 c 2 B' 2 +iS^B's} 
+ {U^+P^BW) +p 2 c 3 B / 2 + p 2 c 2 B / 3+iS 2 c 1 BV; <f+ ... , 
which may be written 
x —X 2 = £1$ + ^ 2 <i 2 + £j< 5 3 + .... 
The formulse which have been obtained avail to determine the displacement, velocity 
and pressure at the shot or its image at any time during the progress of the second 
reflected waves. 
35. General Method for the Second Reflected Waves. —We shall need-to be able to calcu¬ 
late Z, t and x 0 for any simultaneous values of r and s that can occur in the second 
reflected wave from the left. It is best to obtain formulse for t and x 0 separately, and 
not to deduce them from the formula for Z by differentiation, because the formulse 
will be approximate, and to obtain the terms of any particular order in t, for example, 
by differentiation of Z it would be necessary to obtain the terms of order higher by one 
in the formula for Z. The method of determining the formula for x u has been indicated 
already in Article 32, and the work will be completed presently. The formulse for t 
and Z will be found by similar applications of the method of Riemaxn. We begin with 
the formula for t. After finding formulse for x 0 , t, Z, we can calculate x from the equation 
36. Method of Determining t. —The value of t along the locus x {l = 0 has been found 
in Article 33. To obtain the differential coefficients of t along the same locus we have 
the equations 
+ n 7 — = 0 . 
cu OCT 
(^Xq | | ( t _ „ 
a <r du~ ’ 
2 G 2 
