848 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
Again, since we are only seeking to find the secular changes in the ellipticity and 
mean distance, therefore (as before pointed out) we need only multiply together terms 
whose arguments only differ by the lag. Secular inequalities, in the sense in which 
the term is used in the planetary theory, will indeed arise from the cross-multipli¬ 
cation of certain terms of like speeds but of different arguments ,—for example, the 
product of the term F U P 4 E 3 cos (2y— 2/2 — 2f u ) in )£ 3 — &F multiplied by the term 
2P 2 Q 2 J 2 cos (2y — 2I2 / + 2rrr / ) in X' 3 —Y /2 , when added to the similar cross-product in 
4X / Y / 3fT) (which only differs in having sines for cosines) will give a term 
2F H P 6 Q 3 E 3 J 3 cos [2(e r —e) — 2m' —2f H J. This term in the disturbing function will give 
a long inequality, but it is of no present interest. 
The products may now be written down without writing out in full either the 
functions or the X' Y'-Z' functions. In order that the results may form the 
constituent terms of W, the factor ^ is introduced in the first pair of products, the 
factor 2 in the second pair, and the factor f in the last. Then from (280) we have 
v/2_-yv2 V2_ 
2 ; ^j JL ^T L +2X'Y'*© 
=iP 8 {F i E i 2 cos [ (e / — e)-f-(a/ — m) — 2f i ]-f-F“E 3 2 cos [2(e'-e)-2f ii ] 
+ F Ui E 3 2 COS [3 (e' — e) — (m — m) — 2f™] + F iT E 4 3 COS [4(e / -e)-2( CT , -ur)-2f iT ]} 
d-2P 4 (d 4 {FJ 0 2 cos 2f 
+ F‘J 1 2 cos [(e / — e) — (m' — m) — 2f l ]d-F i J 1 2 cos [(e r — e) — (m' — ra^-f-Sf 1 ] 
+ FM 3 2 cos [2(e 7 — e) — 2{m' — m) — 2f ii ]d-F ii J 3 3 cos [2(e'-e) -2( B /-ot) + 24]} 
+1 Q s { FiEj 3 cos [ (e' -— e) + (m — m) + 2fj]+F H E 3 2 cos [2 (e— e) + 2fJ 
+ F Ui E 3 2 COS [3 (e -e) — (m' — m)-{- 2f Hi ] + F iv E/ cos [4 (e — e) — 2 {m— m) + 2f iv ] } 
(281) 
2Y'Z / 15j£ + 2X / Z / £2; = the same, when 2P 6 Q' 2 replaces ^P 8 ; 2P 2 Q' 2 (P 2 —Q 2 ) 2 replaces 
2P i Q i -, 2P 2 Q & replaces ^Q s ; and G’s and g’s replace F’s and 2f’s . (282) 
3 X ,3 +Y /2 —2Z' 2 &+W-2Z 2 
2 3 *3 
= i(P 4 -4P 2 ^+^) 2 {J 0 ^+2H i J 1 2 cos If] 
+2H 11 J 3 3 cos [2(e' —£)“2 (ot / —CT) + 2h u ]} 
+ 3P 4 $ 4 {H i E 1 2 cos [(e'-e) + ( CT , - CT )+h i J-fH ii E 3 2 cos [2(e'-e) + 2h ii ] 
d-H 111 E 3 ~cos[3(e — e) — (m '— <7r)-b3h ul j-bH IV E 4 2 cos[4(e , — e) — 2{m' — m) —J— 4h lv J} (283) 
