193 



C 3 fx A - C . /cos i + 1 



sin i ( — sin (0 - 20) 



dt B 2 r z B 



COS i - 1 . , A n A 



cos i sin 0 + — sm (0 + 20) 



dtv z B - A 3 u ^-i (1 +cosj 2 . . ^ 



1 — COS l ) 



+ J sin 2 i sin 20 + sin (20 + 20\ 



2 ) 



In the first of these three there are three terms on the right-hand side, 

 each of which being integrated will give a term in iv 1 of the general 

 form H sin (x) ; and corresponding to this there will also be a term in w 2 

 of the form iT cos (%). Now, if these two are substituted for and 

 B - A 



tv 2 in the function — — — which occurs on the left side of the 

 C 



third of the above equations, it is evident that they will only produce 

 a periodic term ; but this is because one of them is a sine, and the 

 other a cosine. If, however, they had both of them been sines, or 

 both cosines, the case would have been very different, and the multi- 

 plication of them together would have produced a constant term, 

 "What we have to do, therefore, is to see whether the further develop- 

 ment of the disturbing function will produce any such terms. And it 

 is very readily that it does produce a considerable number of them, cor- 

 responding to different combinations of 0 and 0 of the same kind as 

 those in the equations last formed. Those which I shall select for exa- 

 mination at present are those which have <f> - 0 for their argument ; so 

 that we must develop the disturbing function so as to include all terms 

 of the form sin 0 - 0 or cos 0 - 0 wherever they occur in the first tw° 

 equations ; that is, those for iv x and iv 2 . 



Let us resume, therefore, the two last terms in equation (0) which 

 had been rejected, and which will contain all the terms of lowest di- 

 mensions of the form required, we shall then have for this part of the 

 function on the right side of the equation for 



1 



A r 3 j 2 



- , N r* 15 x (xx x + w, + %% x y 



substituting the values of z, y, &c, in this, and retaining only terms of 

 the form sin 0 - 0, cos 0 - 0, the first part of this expression will give 



for shortness, let 



1 3 m. „ . 1 + cos 1 . " , 

 --^2(^0 — — sin (0-0) 



1 + COS 1 _ 1 - COS « 



— = a and — = /3 



