792 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
Then substituting in (199), we must have 
and 
Whence 
Hence the solution of (199), when A = 0, is 
JO 
X = C0S (a * +,?)_ 4 a(a?-V) Be ^ {a ‘ +v) 
If t be very small, the second of these terms may be neglected. 
By writing y — \tt for rj, we see that a term Bt sin (at-\-y) in the differential 
equation, would have given rise in the solution to 
4,aC(a~ + W)-8a 3 C=B 
2(C—aD)(a i -\-b 2 ) — l‘2a~C-\-4 : a; i D=0 
C-- 
B 
4 a{a 2 -b 2 y 
- 5ffl3 
— J (H/ £-» 7 O\0 
4« 2 (a 2 —& 2 ) 2 
* s * n ( a ^~^~' r l) 
t being very small. 
By this theorem we see that the solutions of the two alternative differential 
equations 
D^F^-HF^ 
are, when t is very small, 
§ K 2 — a :., 2 fz 
z=— , ' . . dFJ ~ l — the same with 2 and 1 interchanged. 
4*i (*i"~* 2 ") Lti 
The similar equations for Db/, DS 7 , may be treated similarly. The general rule 
is that:— 
tz and tt, in the differential equations are reproduced, but with an opposite sign in 
the solution; and similarly ty and ty are reproduced with the opposite sign; and in 
the solution the terms are to be multiplied by 
5 k-, 2 —AV 3 5k„ 3 — K-, 2 
- or 
4/q 2 (/q 2 — /q 2 ) 2 4« 2 3 (/c 2 2 — /q 2 ) 2 
For the purpose of future developments it will be more convenient to write these 
factors in the forms 
"S+ 7 W and 
2/q(/q 2 —/q 2 ) [/q 2 — /q 2 '2k 1 J 2/q(/q 2 — /q 2 ) [/q 2 —/q 
2*2 ■ 1 
2 o l~ r) 
* AW'* ' W - 
