210 



det 2 3 fxA-C . (l+cos< . - — — ) 

 — i + . = - -4 — — sm i — - — sin 0 - 29 - . . . } 

 dt 2 r 3 B (2 I 



If, now, we take the terms already found in the first approximation 

 for i, 0, vyjr, viz., 



. A 1 -B 2 . ■ , A 2 + B x . 



1 = M + i i — sm ft 1 + i — cos ft 1 , &c. 



2 n 1 2 n l 



and substitute them for t and 0, &c, in the above equations, we shall 

 have terms whose argument is n - ft 1 . And it will suffice for the pre- 

 sent if we substitute the values in the first term which occurs on the 

 right-hand side of each of these equations; viz. that depending upon 

 0 - 29 ; for, although that depending upon 0, that is the second, 

 would produce terms of the same form, yet the increment so formed is 

 best considered in conjunction with the term in N which corresponds 

 to it, namely, that depending upon sin 20. Making, therefore, the sub- 

 stitutions mentioned, we shall have to find the increment of u lt 



d» x 3 p, G-B . . /, Ui-Bz . 



— — + .. = — - — ( 1 + cos i - k sin 2 <) < ; — sm n -ft 1 



dt 8 r 4 A y 2 \ ft 1 



A 2 + B, 



COS ft - 



This might be integrated in the same way that the equations for 

 &c, were treated above ; but as it will suffice for the present pur- 

 pose to reject quantities multiplied by products of 



C-B _ C-A 

 — and_, 



we may integrate it without introducing the second differential coeffi- 

 cients. It will give 



3 fi C-B„ . . „ x ( A L -B 2 • 



w i - o "3 — T~ (1 + cos;-ism 2 f)J- 7 -— cosft-ft 1 



8 r 3 A v 2 J \ (n - ft 1 ) n l 



A, + Bo 



+ - ; SUl ft - ft 1 } 



(ft- ft 1 ) ft 1 ) 



which, if we make the same supposition as to i which was made before, 

 becomes 



3 p C-B I A l + B 2 A 2 +B x . 



^1 = T ~T 7— J - == T~T COS ft - ft 1 + -— sm ft - ft 1 



4 r 3 A [ n-n l n l (n- ft 1 ) n x 

 for the increment of u>, above spoken of. 



