THE ELEMENTS OF THE ORBIT OF A SATELLITE. 777 
First take the case (iii.), where the moon is tide-raiser and disturber. Here we may 
take N=N', e=e,j=j' throughout, and after differentiation may drop the accents to 
all the symbols. 
From (129) 
dW It* 
-^7 i /g=-iFi{^cos2f 1 + < ;'cos(i\7— i/r+2f 1 )}= sin (N— xJj) sin 4f x . (152) 
From (130) 
yW /t 2 
-5^/ ¥ = cos gi +i cos (N-xjj-g,)} = sin {N—iff) sin 2 gl . (153) 
From (131) 
/t 2 
~df/ ^ = 2 G i G cos g +i cos (W-i//+g)} = —\j sin (N-xfj) sin 2g . (154) 
Therefore from (152-4) we have for the whole perturbation of the earth, due to 
attraction of the moon on the lunar tides, 
dW 
& T/T=li sin (V— 0)[sin 4f,+ sin 2g,— sin 2g] . . 
(155) 
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 
dW It'* n 
Id / —=0 
di / g 
. (156) 
Next take the cases (v.) and (vi.), where the tide-raiser and disturber are distinct. 
Here we need only consider W 3 . 
From (131) 
f j W Itt' 
-M?j =i G (* cos g+j cos (N-xp+g)} 
When the moon is tide-raiser and sun disturber, this becomes 
~-^j sin sin 2g ........ (157) 
When sun is tide-raiser and moon disturber it becomes zero. 
Then collecting results from (155-7), we have by (18) 
