ON LAGRANGE’S BALLISTIC PROBLEM. 
183 
Along AB we have 
Z = i (c + h) % { ( V 1 _ 1 [ _ 1 (fas), 
a l\^-er 0 + .S/ J a 
where 
so that 
r X AX - Li' 7 " aT'L ■ . .. :fi . L(X o+iii (20 — 18O0 + 42O£ 2 —280£ 3 ), 
05 r + s {r' + s'Y v " ~ 
_ (A^-X^-A) 
(r' + s') (i+i + '^) 
(c + A) 
<Tn 
0 1/ u 0 
9« 
1 x / -4-hi-.-*) 
2 tr ots/ J a 
(K + *0 &-r') 
{r' + s'Y 
20 (!<r 0 + .sf + 180 
;^„-7-0(5-^)(^„+5r 
r' + s' 
Along BC we have 
+ 420 V - K -t g ^^o + - v >') +280 ^°’° r ,^* 
(■ r' + s') 2 {r' + s'f 
ds. 
Z - - AH <r ° 
a 
where 
a I VAo-o + r 
hX - XX = fe . + r ') ( + 2 . -^ 3 ^±!')’ (20— 180f + 420(' 2 —280<; s ), 
dr r + s (r + s ) b 
so that 
0Y 5V X 
? _ _ (+°'o~ g/ ) O’ —^0 
'(,•'+*') (K+O 
z +- 
bc \ or r + s 
dr 
Ho-,, J 
l V^CTq + ? 
i} + f(ko-r) 
(Ao-p + H) (Aq-o—X 
(r' + s') 6 
20 (A+, + >f + 180 X) . X A^)(Xo+ . r)“ 
+ 400 (Aa-p—s') 2 (r-r'Y (A«r„+r) _ (A<r 0 -*T (r-r'f 
+ (ffi + .ffi 2 +- 8U (r' + s'Y 
dr. 
{r' + s'Y 
There is no difficulty in the integrations. After they have been performed, we may 
suppress the accents on r' and s', and so obtain the formula for Z in the first middle 
wave expressed as a function of r and s. We find, for example, as the coefficient of 
(c + //)rr 0 0a the expression 
9 / A q 'o+-X (Aq-p + s) (A q 'o — r ) 
r + s/ \ r + s 
(r + s) 6 
9 f 7 0 4 — 5 (A<t 0 + s) 
-i 8 Ho 4 - 90rr+ (A°"u + s) + 15 (A<r 0 + s) 4 } 
— X°~"+ / " —-X 1189rr,, 4 — 42 O 0 -/ (A+. + *) + 270(T (l 2 (A cr o + ,<? )“ —35 (Ao-o + s) 4 } 
(r + s) 8 
- {l 26 o- 0 4 — 420 o - 0 3 (A^u + s) + 540 (r 0 2 (A+Ws) 2 - 315 o - 0 (A«t 0 + s) 3 
+ 70 (As-o + s) 4 }• 
2 D 
VOL. CCXXII.-A. 
