THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
849 
The sum of these three last expressions (281-3) when multiplied by 
equal to W the disturbing function. 
o 1 
T 1 
3 (l-e s )“ 
§ 24. Secular changes in eccentricity and mean distance. 
Before proceeding to the differentiation of W, it is well to note the following coin¬ 
cidences between the coefficients and arguments, viz. : E^ occurs with (e' — e)-j-(nf—<rr), 
E 2 2 with 2(e / —e), E 3 3 with 3(e'—e) — (to' —ut), E/ with 4(e 7 — e) — 2 (ct'— «t), J, 3 with 
(e'—e) — (trd — rS), J 2 3 with 2(e'— e) — 2(rn / — ttt), and the terms in J 0 3 do not involve e, e, 
ct, is. In consequence of these coincidences it will be possible to arrange the results 
in a highly symmetrical form. 
By equations (11) and (12) 
~l l lo ^G£,+yjf)w, when y=\ 
and 
1 dP (d , d \„ T . 
k * = wIiea ^=° 
Hence the single operation d/de -\-yd/dis will enable us by proper choice of the 
value of y to find eit her £d log y/hdt or d£/Jcdt. 
Perform this operation; then putting e' — e, ur' = <rr, and collecting the terms accord¬ 
ing to their respective E’s and Js, we have 
fdW dW\ m t 2 1 
\de' +y dw')~§ (1 —e 2 ) 6 
=E 1 3 (l-by){i|P 8 F 1 sin 2f 1 + 2P°() 3 G 1 sin g 1 —2P 3 $ 6 Gi sin gi—sin 2fi 
— SP^fH sin If) 
+ E 2 3 (2){the same with ii for i, and 2h“ for h 1 } 
+ E 3 2 (3—y){the same with iii for i, and 3h m for h 1 } 
+ E 1 2 (4 — 2y) {the same with iv for i, and 4h lv for h 1 } 
+Ji 2 (l — y){2P 4 Q 4 (F I sin 2f i -~F i sin 2f 4 ) + 2P 3 $ 3 (P 3 ~() 3 ) 2 (G i sin g ; —-Gisin g ; ) 
-i(P 4 -4P 3 g 2 +G 4 ) 3 H ; sin If} 
-j-J 3 3 (2 —2y){the same with ii for i, and 2h u for h 1 } . . . . . . . (284) 
The functions of P and Q, which appear here, will occur hereafter so frequently that 
it will be convenient to adopt an abridged notation for them. Let x then represent 
either i, ii, iii or iv, and let 
