ON LAGRANGE’S BALLISTIC PROBLEM. 
‘213 
We require also the differential coefficient dZ'/ds along the locus, and this is given 
by the equation 
= —t— — (T 2 + Ci<5 + C 2 cP+ ...). 
cs 
The value of Z' at any point ( rs') is then given by the equation 
Z' K .0 = [VZ'L + J (f - £) Z' *•+ (f+f) V *, 
in which V has the same form as in Article 36, the value of Z! at A is 
diS A + d 2 S A + d 3 S A 3 +..., 
and the integration is taken along the locus. 
The result may be recorded in a similar form to that for t' in Article 37. The terms 
of the first order in the formula for Z! are 
the terms of the second order are 
/ 1 + e + $ A \ 5 i ? 2 1 
^) dA 2 - 
\l+e + Sj W2 " A (l+e + <i ) 5 
the terms of the third order are 
{P 2 { Cl + 5T 2 (B ' 1 + l)} -K.} 
yyyyy j d 3 S A — yyyyyy, [ 3^3 {c 2 + 5(+ (B'i +1) + 10T 2 (Bh +1) 3 + 5T2B' 2 } 
—-f {d 2 + 4ch (Bh + l)}] 
+ 
10 
10 
(l+e + J) 6 3 
and the terms of the fourth order are 
(S 2 T 2-dJ B\ (S + 2S a ) {S-SJ - , S 2 T 2 e ($-SaY, 
(l+e + <5 ) 6 
'AtsY 1a ‘” (TT7M?’ [i2s {C3+ 5ca (B ' I+1} + 10Cl <B '' +1 ) 2+5Cl]B ' a 
+ 10 T,(B' 1 -H) , + 20 T,(B ' 1 + 1)B',+5T,B' !> } 
- f { d, + id, (B', + 1 ) + 6 d, (B', + 1 f + id, B' a }] (S'- V) 
- (1 + * , [VS. {«i■+ 4T„ (B\ + 1 )} + W (BV-5)] e (S + 2 i A ) (i■-4) a 
+ —-— 7 [|S 3 {ciB , i + 4T a B'i (B'i + 1) +'I 2 B'j} — 3 ! '-4li\ di B: (3+ 9) -t- 2 c?,B 2 }J 
( 1+6 + 0 / 
x(^+ 2 W + 3i i a )(i-i i ) ! . 
