724 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
Then let x=DA, and we have 
x -b=I)C = x / 
Again, if M be Diana in her orbit, MB = /, and since MG=/ /} therefore 
/ + a=+ 
Whence 
cos x , = COS X cos b + sin x sin b 
sin x, = sin x cos b — cos X sin b 
cos l — cos l cos a— sin / sin a 
✓ 
sin / = sin 1 cos a + cos l sin a 
Substituting these values in the first of (19) we have 
Mi = cos x cos l (cos a cos b — sin a sin b cos i )+sin x cos l (cos a sin b + sin a cos b cos i f ) 
— cos x sin l (sin a cos b+cos a sin b cos i) —sin x sin l (sin a sin b — cos a cos b cos i) 
Now cos i t — —cos C, and 
cos a cos b + sin a sin b cos C = cos c=cos N 
cos a sin b — sin a cos b cos C = sin a [cot a sin b — cos b cos C] = sin a cot A sin C 
= cos i sin JY 
sin a cos b—cos a sin b cos C=sin b [cot b sin a—cos a cos CJ=sin b cot B sin C 
= cos j sin N 
sin a sin b+cos a cos b cos C = sin a sin b + cos c cos C — sin a sin b cos 2 C 
=sin a sin b sin 2 C + cos c( — cos A cos B + sin A sin B cos c) 
= sin A sin B sin ' 3 c+sin A sin B cos 2 c—cos A cos B cos c 
=sin i sin j —cos i cos j cos N 
Then substituting in the expression for M 1? 
Mi=cos x cos l cos A+sin x cos l sin A+os i — cos x sin l sin N cos j 
— sin x sin l (sin i sin j —cos i cos j cos N) 
Let P =cos Q —sin p —cos \j, < 7 =sin \] 
Then 
Mj = (P 2 + Q-) ( p~-\-q~) cos x cos l cos A 7 + (P~ — Q~) (p~ + (p) sin X cos l sin N 
— ( P 2 + $ 2 ) ( p 1 — <p) cos x sin / sin N + [P l — Q~) (p~ — q~) sin x sin 1 cos JV" 
— 4 PQpq sin x sin l 
— P-p 1 cos (x — l—N)-\-P' : q~ cos (x + ?— N) J rQ z p 2 cos (x+?+A) 
+ Q 2 q~ cos (x— l+N) + 2PQpq [cos ( x + l) — cos ( X —0] • ( 20 ) 
