ON LAGRANGE’S BALLISTIC PROBLEM. 
203 
lable to three. This difficulty was found to present itself in the calculation of the second 
reflected waves by this method, and another method had to be sought. An account 
of this will be given in the theory of the second reflected waves. 
29. Pistons of Equal Mass .—A considerable reduction in the number of coefficients 
to be calculated is effected by supposing the two pistons to have the same mass. When 
this is so H = h, and hereafter we shall write everywhere h for H. The calculation of 
Ay, A 1? ... , A 5 is then simplified a good deal. Further, it appears that the coefficients 
a differ only in sign from the coefficients A, or we have 
a n = 
■A 
0> 
a, 
= -A 
1? 
It is now unnecessary to calculate separately the pressures, velocities, displacements, 
and times at the two pistons. We shall speak of the piston specified by x 0 = c as the 
shot,” and of the piston specified by x 0 = 0 as the “ image of the shot.” AVe shall 
generally calculate the pressures, &c., for the image of the shot, because a slight simpli¬ 
fication is effected by putting x 0 equal to zero. 
30. Incidence of the Second Middle Wave upon the Pistons .—The value of s at the 
receding front of the second middle wave is that which has been denoted by s 1: and 
in the case of equal pistons it is the same as R : or —|-cr 0 . This is therefore the value 
of s at the image of the shot at the instant when the receding front of the second middle 
wave reaches it. It will be denoted by S 2 . The corresponding value of r may be found 
from the formula 
r-R, = Bj (s-S l ) + (B.Jt 1 )(s-S l Y + ... 
by putting S 2 for s. It will be denoted by R,. From this the corresponding value of 
o- may be found. It mil be denoted by S 2 . The corresponding value of u, which is 
R 2 —S 2 , will be denoted by U 2 . The corresponding value of Z, denoted by Z 2 , can be 
found most simply from the formula for the first reflected wave from the left. We 
have 
Z 3 = K 1 + L 1 U 2 +105 Fl ( 2 fi) -105 Fl<1 ^ R2 ) +45 - X0 
^2 ^2 ^2 
V 5 
where 
F t (2R 2 ) /2A 10 f Ao _ Aa / 2 Ri —2R 2 \ 
2 2 1u \tj XffV Si / 
1 2!_A 2 / 2R 1 -2R a y 
2! S x 8 V ^ ) 
F 1 ‘ 1> (2R S ) 
AiY. 
fA, 
2! A., /2R ; -2Ro\ 
S' 9 
w 2 
w 
U, 9 
Si® \ S x ) 
1 3! A s /2R X — 2R 2 \ 2 
+ 2! td V Si ) 
1 3! A,/2R 1 -2R 2 \ 3 
1 4!A 4 / 2Rx- 2R 2 \ 3 
3! Si 6 \ Si / 
J 
