ON LAGRANGE’S BALLISTIC PROBLEM. 
185 
while for v — 9 we have 
/'! _o V j ffiT 1 
\<T 007 l O’ J 
( 2 . 4.6 
8 ) -C£ 
< 
rx 
) 9 l 
T 
and for c = 10 we have 
/I 3 \ 4 f (a- + -M) 10 l 
\<T d<T/ 1 iT j 
2rt (2. 4 .6.8 . 10) + 
It follows that the expression for Z can be written either in the form 
or in the form 
£f 
Qi (o- + w)\ 
7 J 
+ lv T + L x u, 
z = (-JLV |2iLr!41 + k,+ku, 
\(T per/ [ o- J 
where 
K, = 2 .(2. 4 .6.8) x the coefficient of (<r—u ) 9 in dq (c r—u), 
Lj = —2 . (2. 4 .6.8 .10) x the coefficient of (0-— u ) 10 in dq (<r—u), 
\ = 2 .(2. 4 .6.8) x the coefficient of (a- + u ) 9 in ffi, (<r + u), 
b = 2 .(2 . 4 .6.8.10 ) x the coefficient of (o- + u) w in ffi, (a- + v), 
and Qt and q x are certain rational integral functions of the 9 th degree. The explicit 
expressions are 
Ki = — ^ (c + h) (o-Ja), L, = —lila, - gH <rja, h = — H/a, 
+ ~ 945 ~x 2 ^ ] ( 315<r ° 8 ^ + u )-^^o{cr + uy + S 7 S ( r 0 i (<T + uy 
— 180 o - 0 2 (f r + n)‘ + 35 (<r + u) 9 }, 
and q x (c r—u) is obtained from Qj (ar + 11) by writing — (c r—u) for (er + u). 
17 . Incidence of the First Middle Wave on the Pistons. —The values of all the quantities 
at the piston M, at the instant when the first middle wave reaches it, are to be found 
from the formulae, connected with the receding front of the wave, by putting x 0 = 0. 
We see that the receding front reaches the piston M at the time T 1} where 
T _ (h+W _ 1 
«H a ’ 
that the corresponding value of o- is 2 l5 where 
