THE ELEMENTS OF THE ORBTT OF A SATELLITE. 
801 
4- ^=t~ 1 i -U -P)-a:h Kl + u -Vz, 
L x dt (/cj—*jj) s [ v 1 A H ’ /c 2 + « 
T { 7 (^ 2 +“) + §(«:!+a)+gb da} 
(k 1 —k 2 ) 
1 dL 1 _ _1_ 
Z 2 dt (/q—/e 2 ) 2 
1 
/q— /Cg 
1 
— (^2 + a ) K — — a ' b ^^ — b ' a 
{y( K i + a ) + §(* 2 + a ) +gb — da] 
1 ZZf _ 
A' dt ^K -« 2 ) 2 
1 
— (^o + a)(a / — /S') — a'b—b r a 
/q + « 
+ {7( ,c 2+ a ) + §( K: id _a )d - S b — ba l 
/q—/c 2 
1 
1 dLJ _ 
Z 2 ' dt (/q — /c 2 ) 
1 
/q —/Cg 
i] -(* 1 +«)(«'-/8')- a 'b-b'a^±; 
{y(/Cl + a) + S(K : 2+a) + gb —da} 
(224) 
We shall now show that these four equations are equivalent to two only, and in 
showing this shall verify the correctness of the results. 
To prove that the four equations (224) are equivalent to two. 
In (118) we showed that 
Zfi_ /q + < 
Z, — a 
Therefore we ouerht to find that 
1 dLf 1 df_ 1_ d_. v_a' 
Zfi cZ£ Zj dt /q + a dt' 1 a 
k.+/ 3 d, , \ a' 
:fyi+ a )-r 
ab Zf 
Now by (188) 
and 
so that 
2(/c 1 4' a ) = a —/S— fy («—/3) 2 + 4ab 
0 A . , , (a-/3)(a'-/3') + 2(a'b + ab') 
y K >+“)=“ -^+- 
d, . («'—/30(/q + «) + a'b + ab' 
-fyi + a ) = 
cK 
/Cj /Cg 
And thus we ought to find that, 
. . T 1 dL{ 1 dL x 
(^■^Lzfy dt~T,Ht J 
bh 
= a' —/3' —-(/c ; + a)—r-(/c 3 +a) 
