ON LAGRANGE’S BALLISTIC PROBLEM. 
187 
for nufct. Then we have 
M 9^ 0 _ _i (T \ u / 9>r 0 dl 9a' 0 dt \ 
o»p 0 9(7 \(r 0 y \9<r 9 u 9 u 9or / 
Again we substitute — II 9Z/9 a for x 0 and 9Z/9 u for t, and put 9Z/9 <r — 0, obtaining 
the equation 
9^Z ffiZ _ / 9 2 Z \ 2 = _ 10IW 0 9dZ. 
9(7 2 9 U 2 \dcrdvj a<r u 9o- 2 ' 
The condition which holds at the junction is that 
Z — V J kb (q - + U 
\rr daj 1 (7 
)l 
+ + LjU 
for all values of <r and u for which o- + u = 2R : . 
19. Determination of the First Reflected Wave from the Left .—These conditions can be 
satisfied by assuming for Z the form 
Z = f-+ 
\(7 err 
i 9 Y/Fik+y) 
+ Ki + Lj u, 
<7 
expanding the unknown function F, in the series 
Fi (cr+w) — A 0 + Aj (cr + u —2li,) +A 2 (rr+u — 21k) 2 + ..., 
and finding the coefficients of this series. 
The condition which holds at the junction determines the coefficients A 0 , A,, ..., A 4 . 
The condition that 9Z/9o- vanishes when r = lb and s = S : determines the coefficient A ,. 
The remaining coefficients are to be determined by the condition which holds at the 
piston. 
We have 
Z = 105 
F,0+«) tu PWO+O , , UV JI 0+«) 
45 
-10 
F, (a (<r + «) ^ F,«U+o) , J, 
+ Kj + LiU, 
from which we find 
K = 0,(28,), A, = 0«'(2K,), 2! A, = Q, I2I (2K,). 31 A,- Q,«(2B,), 4! A, = Q,»(2B,). 
We have also 
9Z q . r 
= —94o 
Ul±u) +!)45 F»'(^ + m) „,,-, Fi a O + ») 
10 
420- 
cr 
cr 
+ 105 
F, B) Q + «) _ Wm) 
cr" 
(7 
or 
