THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
779 
From (131) 
dW 0 /r 2 
—j -— —|G{r sin g+ij sin (N—xjj+g)—ij sin (N— xjj— g)+ p sin g} 
d X / 0 
dW 2 /r>_ 
~df'/ g“ 
2 ^W | /t 2 _ 
<¥/ s 
—y sin (JV— </>—g)+/sing} 
(164) 
Therefore 
% 
;= -<?&{?' sin g+i sin {N-xp+g)} 
d X '~i dir']/ 
= — cos (iV— xjj)) sin 2g .... (165) 
Then collecting results from (161-3-5), we have for the whole perturbation of the 
earth clue to the attraction of the moon on the lunar tides, 
"1 dW 1 dW 
d/i _ iy2\_. _± 
A 2 ’ dy/ i dxjr'_ 
/——it; 
/ 5 
— iii+J cos (N — xjj ))(sin Ifj-j-sin 2g x — sin 2g) . (166) 
The result for case (iv.), where the sun is both tide-raiser and disturber, may be 
written down by symmetry ; and since j— 0 here, therefore 
1 , dW 1 dW 
yl 1- 2 l /d x '~i df'_ 
/y=i* sin 4f 
. . . (167) 
It is here assumed that the solar slow diurnal tide has the same lag as the sidereal 
diurnal tide, and that the solar slow semi-diurnal tide has the same lag as the sidereal 
semi-diurnal tide. This is very nearly true, because fl' is small compared with n. 
Next take the cases (v.) and (vi.), where the tide-raiser and disturber are distinct. 
Here we need only consider W 2 
dW % /tt' 
d x I g" 
■ \ G {v sin g +ij sin (N — xjj +g) ■— if sin ( N' — xjj' — g) 
+jj' sin N ' +g) 1 • ( 16 8 ) 
d W 2 jrr‘ 
df 7 0 
- 
■if sin (A 7/ ~xjj r ~ g)sin (N—N'+g)} 
U 2 dW 3 /yrj_ ( 
hi 
d X / 0 
