THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
809 
In the last section we have determined the functions a, a, &c., and have them in 
such a form that F, G, A, D (or y, g, §, cl) have all a common factor d^/nkdt. 
But this is the expression by which we have to divide the equations in order to 
change the variable. 
Therefore in computing r, G, &c. (or y, g, &c,), we may drop this common factor. 
Again cl, a, /3, b were so written as all to have a common factor rt/n ; therefore 
k 1 and k, 2 also have the same common factor. 
Also cl a', (3', b' all have a common factor ( dg/kdt)(Tt/n~ ). 
From this it follows that when the variable is changed, we may drop the factor rt/n 
from cl, a, (3, b, k v k 2 and the factor (d£/kdt)(rt/n°) from a!, a r , (3\ b'. 
Hence the differential equations with the new variable become 
hi — log tan \ J = —; 
(«! — rc 2 y 
— (k 1 + cl) (cl — (3') — ah 
n K \ "t" a 
K.i + U 
■b'a 
cl 
log tan II = 
+ ^ ^ Ot) -hgb — da} 
f _ (^ q_ a )( a ' _- ffb - b'a^ 
(/q — k 3 ) 
{y(^i + a) + S(/c, + a)+gb —cla} 
J 
(250) 
or similar equations with r, G, A, D in place of y, g, S, d if the viscosity be not small. 
But we now have by (232-3-5-6-7-9, 242—3-4—5—6—7) 
a =m + 
cl = in 
t' i 
t 2Xc’ 
H _3_ 
t 2Xc 
a=m, 
/3 —-Iff- , b — 1 
1 
1 + H +7m I , a'=-m 2 i+ - +7m 
/3' = — j i+ T - + f-V +(- V"+6in I , b'= — ji + G )+<3m 
r _ i m sin 4f i~ sin 2gi + sin 2 g 
2 sin 4f x 
(sin 4f x + sin 2g x — sin 2g) — 2~ sin 2g + ( - ) sin 4f 
A=_-_ - _ 
2 sin 4f, 
y • ( 251 ) 
m 
7= 
T T 
1-2X— + - 
T \ T 
2(1—xy 
2(1-X) 
2 1+ — sin 2g —2 sin 2g x 
bG-aD=Hn V T 
sin 4^ 
m 2X+- 
bg— acl= —a 
2(1—X) 
