ON LAGRANGE’S BALLISTIC PROBLEM. 
1 80 
At the junction, where s = s u the value of Z must be the same whether it is found 
from the formulae belonging to the reflected wave or from those belonging to the first 
middle wave. At = c the equation of motion of the piston, viz., the equation 
dll 
m — = orp 
must hold. The equation — 11 cZfdcr = c must hold at r = r, and s = s\. 
To express these conditions it is convenient to write 
then Z' satisfies the same differential equation as Z, and cZ'/co- vanishes when x t , = c. 
The condition which holds at the junction is that 
z = (- ff +kl+ i,u 
\<T dcr / l cr J 
for all values of a and u for which cr — u = 2sj. 
The condition which holds at the piston is that 
FZ' _ / d 2 Z' Y = lOAo-,, 10 FZ' 
dcr 2 du 2 dcr on) ac r 11 dcr " 
when dZ'/dtr — 0. 
These conditions can be satisfied by assuming for Z the form 
z = (I A 4 
1 ! cor,, 11 , v 
— f ----1- /, ( cr — u) 
cr j 945a J V 
+ k\ + k i'll, 
expanding the unknown function /j in the series 
f\ — a 0 + a x (cr — u — 2+) + a 2 (cr — u — '2s x ) 2 + .... 
and finding the coefficients of this series. 
For the coefficients a 0 , o 1} ..., we find 
®o + 
Cc r, 
Hi 
945a 
= q, (2s : ), a x = q 1 (1, (2.? 1 ), 2! a 2 = q 1 (a) (2s 1 ), 3! a 3 = q 1 (3) (2.sq), 4! a 4 = (2s,). 
The coefficient a, is given by the equation 
a„ 
- 945 ^+ 945^-420 
a 
Q * 
O’, 
2 + 105 3 -15 
4! a 4 5 ! a- 
— + -A = 0. 
a, 
cr, 
cr, 
cr, 
The remaining coefficients can be determined from the condition which holds at the 
piston in the same way as the corresponding coefficients in the formula belonging to 
the first reflected wave from the left could be determined. 
