ON LAGRANGE’S BALLISTIC PROBLEM. 
193 
and for s —s' we have to put t 1 (S—S > ), where S' stands for (s' — SJ/Sj. If the expan¬ 
sions are carried as far as the fourth order, the result is that, to the fifth order in S', 
(Y /<r)du = (Bj — 1) <$'+ {B 2 — (3Bj — 2) (Bj — 1)} S' 2 
AC 
+ (B 3 -3B 2 (2B 1 -1) + (7B 1 2 -6B 1 + 2)(B 1 -1)} J' 3 
+ {B 4 -2B 3 (3B 1 -l)-3B/+|-B 2 (42B 1 2 -33B 1 + 5) 
-|(28B 1 8 -21B 1 2 + 9B 1 -2)(B 1 -1)}<5' 4 
+ {B 5 -iB 4 (30B 1 -7)-6B 2 B 3 + T L r B 3 (210B 1 2 -108B 1 + 7) 
+ £B 2 2 (105B 4 -27) -lYB, (560B 4 3 -462B 4 2 + 81B, - 7) 
+ H126B 1 4 -56B 1 3 + 21B 1 2 -6B 1 +1)(B 1 -1 )} S' 5 . 
Now at any point (R x , s') on r = R 4 the formula for the first middle wave gives 
X a = 
+ Kj + L {ii 
a 
cr„ 
■945Q, (2Rj) +9450-Q/ 13 (2R 4 ) -420cr 2 Q 1 (3) (2R 4 ) + 105o- 3 Q 4 (3) (2R,) 
-1(2R,) +o- 5 Q 1 (5) (2R 4 )}, 
in which we have to put cr = ^ 1 (1+^'). Then, since x 0 vanishes with S', we have 
without any approximation 
*0 = - ^ [{945Z 1 Q 1 (1) (2R,) -8402^ ® (2R>) + 3152 1 3 Q 1 (3) (2R 4 ) -60^ 1 4 Q 1 (4) .(2B 1 ) 
+ 52 1 5 Q 1 (5) (2B 1 )} S' 
+ { — 4202 1 2 Q 1 (2) (2R 1 ) + 3152 1 3 Q 1 (3) (2B 1 ) — 902 1 4 Q 1 (4) (2R 1 ) +102 1 5 Q 1 (5) (2R 1 )} S' 2 
+ {1052 i 3 Q 1 (3) (2R 1 ) -602 1 4 Q 1 (4) (2R 1 ) + 102 1 5 Q 1 (5) (2R 1 )} <i' 3 
+ { -152 1 4 Q«(2R 1 ) + 52 1 5 Q 1 (5) (2R 1 )} «i' 4 
+ 2 1 5 Q 1 (5) (2R 1 )r], 
The coefficients B 1} ... , B, can be determined successively by equating the 
coefficients of powers of S' in the expressions for I (Y fa) du and —x 0 /5H. If 
Jac 
additional coefficients B 6 , , are desired, they may be found by equating to zero 
the coefficients of powers of S' higher than the fifth in the expansion of 
(Y fa) du. 
Jac 
The expansion of x 0 (R 4 , s') in powers of S' may, of course, be found from the expression 
for Z in the first middle wave without transformation to the Q form. In particular it 
may be proved that B 4 = 6 — 4S 4 /or 0 . 
24. Second Method of determining the Coefficients A. —When the coefficients B are 
known, the coefficients A 6 , ... may be found in the following way. 
2 e 2 
