ON LAGRANGE’S BALLISTIC PROBLEM. 
181 
On elimination of dx 0 there results the equation 
2Udt+(t+^jdU = 0, 
which can be integrated in the form 
(t + — ) II = const. 
\ a J 
To determine the constant there is the condition that when x 0 = \c and t — \cja, the 
value of II is a. Hence at the receding front we have 
(ai + H) 2 ri = (!c + H) 2 a, 
or 
(x 0 + H)(^ + H) = (ic + H) 2 . 
13. Conditions satisfied at the Advancing Front. —At the advancing front we have 
in like manner 
/ h 
dx 0 —lldt — 0 and c + h—x 0 — II it + — ) = 0 , 
a! 
leading to 
{at + Kf\\ = {fic + hf a 
and 
(c + h—x 0 ) (at + h ) = {fie + h ) 2 . 
14. Conditions determining the First Middle Wave. —It will now be convenient to 
restrict the value of n to be an integer. This happens when y has one of the values 
3, 5/3, 7/5, 9/7, 11/9, . . . With a view to applications, in which the value of y 
is 1-2 nearly, we shall take the value 11/9 for y, or 5 for n. Then in any compound 
wave Z has the form 
(l AY J *'(* + «)+f(<r-u) \ 
V 3c r) 1 (T j 
or 
105cr _9 F (a + u) — 105cr _8 F (I) (cr + u) + 45<r“ 7 F (2> (cr + u) — lOcr -6 F (3) (cr + u) + cr^ 5 F <4) (tr + u) 
+ 105o- _9 /(cr — u)— 105o- _8 / (1) (rr — u) + 45cr“ 7 ,/ ( " ) (cr — u) — lOcr -6 / (3) (cr — ?/) + cr _5 / (4) (<r — u), 
where F (1) , F (2) , and so on stand for the first, second, &c., differential coefficients of 
the function F with respect to its argument. We have to determine the unknown 
functions from the values of Z at the advancing and receding fronts. 
At the advancing front, where r = -|o- 0 and o- = |cr u -)- s, we have 
