76 2 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
These equations suggest the solutions 
z=%L cos (Kt-\-m) cos (/c^+m) 
y —SL sin (xt + m) rj = %L' sin (kI + m) 
Then substituting in (116), we must have 
— Lk=<xL-{-&L' ; —L , K=/3L / -\-hL 
Wherefore 
_Z7 K + a. b 
L a k + /3 
and 
(/c+ a )(«+/3) — ab = 0 or ht -\- k(cl-\- (3) afi— ab=0. 
This quadratic equation has two real roots (/q and /q suppose), because 
(a-f /3) 3 — 4 (a/3—ab) = (a —/3) 2 +4ab is essentially positive. 
Then let 
K \ "b k -2 — ~ ( a + ft) 
/q— K *—— {(a—yS)~ + 4ab}“ 
And the solution is 
^ sin 2 j cos N=z = L 1 cos (qi-j-nq) d-Z^ cos (/c 2 i+m 2 ) 
sin 2 j sin N=y=L l sin (/qtf+nrJ + Z^ sin (/c 2 i-fm 2 ) 
^ sin 2 i cos xjj ~£ ) =L l ' cos (K 1 £+m 1 )+Z 3 / cos m 2 ) 
^ sin 2 i sin xp = y=L l ' sin (/q^d-m^d-Zd sin (/c 2 ^+m 2 ) 
where 
Ly Ky “J - U b Tj o /Cg a b 
Ly a icy + J3 ’ L 2 a « g + (3 
From these equations we have 
37 sin 3 2j=L 1 2j rL 2 2 -\-2L l L. z cos [(/c x — /q)£-j- m i — *%] 
\ sin 2 2i=L{-\-L 2 -\-2L{L.7 cos [(#c x —/c 2 )^+m 1 —m 2 ] 
From this we see that sin 2 j oscillates between 2 (Ly-\-L. 2 ) and 2(Z^Zo), and sin 2 i 
between 2(Z/d-Z/) and 2(Z 1 , -*~Z 2 / ). 
Let us change the constants introduced by integration, and write X 1 =^sin2; 0 , 
L i—l sin 2 i 0 . 
Then our solution is 
• • (H8) 
