ON LAGRANGE’S BALLISTIC PROBLEM. 
179 
or 
x = x 0 + 
dt = ay + 
Zola 
J '<T 
(< T o-o-) 
<Z0 
jU 
x 0 +H 
a 
_ 
0-0 
N7TI aor > 
0?o 
x = x 0 + — (ay + H) 
ct 
2 n 
2n -1 
This equation holds so long as the plane of particles is in the region occupied by the 
progressive wave. In particular, the displacement of the piston M is given by the 
equation 
2 n 
2 71 — 
I \ H /' 2 n 
[^,-rr) — + I “ 
a 
2n — \ 
cr-cr,, 11, 
in which 
a 
Tlie corresponding values of Z are found from the formula 
in which 
to be 
Z 
dx 
OXa 
+ ut—x, 
( •*-' _ A) ft pll p 
0a* o p ’ p 0 ft—p 
H ( 2 n _ <t (I 
a \2n— 1 fT " 2n— 1 
11. The Progressive Wave from the Eight. —The equation of motion of the piston 
m is 
cu 
m — = to p, 
ot 
and we put 
h = mn- ll a/2u(,g) ir 
Since in the progressive wave r is constant and equal to or o- + u — o- 0 , the values 
of or and / at 2 y = c are found to be connected by the equation 
and then the progressive wave formula is found to be 
at + h _ / ajV” _ 
c + h —ay \<t! 
The value of x for any plane of particles specified by a value of ay in the interval 
h c < < c, and for any time later than that given by t = (c — ay)/a, is found to be 
given by the equation 
