ON LAGRANGE’S BALLISTIC PROBLEM. 
195 
The equation for A 8 is 
8! A s 
3! A, 
4Bi 3 ^8 = 46,3/945 Ai_f ^_945 iIAi +4 20 
5! A s inc 6! A, , 1C 7! A 7 \ 
2 5 
■105 
2 ‘ +i5 ^v 
+ 6B 1 2 (B 1 + l)(-945 ?iAi+840RA-315^# ; + 60iiA-5B4l' 
\ ^1 ^1 ^1 / 
+ 6B 1 (B 1 +1) 3 (420 
^ ^ -315 4 : ^ 4 _l an ^ ‘ _ i n (j • 
+ 90 
V 5 
-10 
V 4 
+ 3 (B 4 + l) 3 ( -105 44+60 
4 • i _ y q 5 ! A,^ 
V « 
n 
V 5 / 
«] / 
+ 6B 1 B 2 (945%4- 2 -1050^2 + 525^4- 4 -150^^ + 25^4 l6 -2^A 7 
■^i . 
5* 8 
- 1 ! 
3! A, 
+ 3(B 1 +1)B 2 (-1050^A 2 + 1050 
■450 
4! A 4 , 4 AA 5 ! A, , 6! A, 
? e 
100 
S' 5 
-'1 
10 
V 4 
^1 
+ 3B 3 ( 945— —1050 + 525 
\ -'I —a' 
3| A i ,_ 150 4LA 4 iOS 5!A 5 
"l 6 
+ 25 
S' 5 
-n 
_o 6 • Ac 
V 4 
^1 
The equation for A 9 is 
2Bj 
4 9! An 
2Bj 4 945 
4! A 4 
S' 6 
^1 
— —945 
5! A 
+ 4B 1 3 (B 1 + 1) ( -945 
V 5 
■+ 
4! A 4 
S' 6 
- +420 
+ 840 
6! A„ 
S' 4 
■*1 
5! A s 
S' 5 
^1 
5-105 A+ + 15 8!A, 
-315 
6! A 6 
S' 4 
^1 
+ 60 
7! A 7 
s' 3 
-5 
8! A, \ 
>► 2 / 
^1 / 
+ 6B 1 2 (B ] + J) 2 (420 
ilA_3i5 44 + 90 44-10 ' 1 
—'i 
S' 4 
^1 
S' 3 
^1 
+ 6B, (B, +1 ) 3 ( -105+A + 60 AA _ ! 0 51A 
*1 
s' 4 
■'l 
+ 6 (B 1 +l) 4 ( 15 44-5 5! A - 5 
S' 8 
— I 1 
S' 5 
■*1 
+ 12B 4 2 B J945 2^ 3 - 945 ^5^ + 420- 105 ^4^ +15 ^nr - 
\ 
*1 / 
+ 6B 1 .(3B 1 + 2) B 2 (-945'^4^ + 840^4- 4 -3 1 5^r^+60^4 :S -5^4 Z ) 
.^i / 
V 5 
-'1 
+ 6 (Bj +1) (3B 4 + l) B 2 (420 
+ 3 (B 4 + l) 2 B 2 ( -315 
3 ! A a c 1 K 4 ! A 4 
_ -315 
V 7 V 8 
^1 -<1 
+ 90 44-10 ^^8 
< 5 
^1 
V 4 
^1 
3! A 3 
■ S' 7 
“*1 
f 180 
4! A< 
S' 6 
^1 
•30 
5! AA 
s' 5 j 
«1 / 
+ 3B 2 2 ( 420 — 525 -^4+300 44 ~ 100 44 +20 44 ~ 2 44) 
w i ^i -n -a —1 1 
+ 66^3 (94544-105o44+52544-150 
S' 8 
-O 
S* 7 
S' 6 
-'l 
5! A. 6! A, .-,7! A, 
S' 5 S' 4 •" S' 3 
*"1 ~<1 —1 
+3 (B t +1) b 3 (-1050 44+ 1050 44 - 450 44+ 100 44 - 11 ° 44 
5 "1 ^1 ^1 ^1 ^1 
f n 1 r A 4 inf;A 2! A 2 ^ 3 ! A s i ka 4 ! A 4 , OR 5 ! A 5 0 6 ! A H ^ 
+ 3B 4 945 -1050 
S' 9 
1 
S' 8 
^1 
s' 7 
— 1 
- —150 
S' 8 
^1 
25 
S' 5 2 S' 4 i 
*-n —j / 
The method avails for the calculation of as many coefficients as may be desired 
