208 



If now we put for 2 . and 2 X their values, and multiply this by the 

 value found above for 1 + cos . t 2 , we shall have, observing that the 



constant terms in 22^ cos 2n - 2n l destroy each other, and retaining 

 the constant term, 



/, no • 772 7Tn if 1 COS < x - l 2 COS + l 2 1 



(1 + cos sm 20 - 20 = j ( - - + - 



1 r~ \ 1 1 



sin 2 l x - - 1 + cos «i cos l x = — - - 1 + cos a — — rr~- . > P 



4 )2n-n x n x 8 (2»- rc) (n- n') n l ) 



the former part may be put into the form 



^ - £ sin 2 < - ^ 1 + cos < 2 cos <] 



or 



- J sin 2 < - J 4 cos « (1 + cos i x - 1 - cos i x ) or - J 1 + cos i 

 therefore the whole expression becomes 



1 + cos t x 



n 1 



(2n - n 1 ) (n - n l ) n 1 2n - n l n 1 



- 4- 1 + cos <, 2 P. 



(n - n 1 ) n 1 



Therefore, finally, the function which we have been examining contains 

 the constant term 



fB - A\ a 3 1 -= „ 1 



^ - - 1 + cos , 2 — (A x + B 2 A i + B l -A l - £ 2 A % - B x ) 



V C J r 3 8 8 ■ (n- n 1 ) n 

 or, as it may be more shortly written 



B-A\ 3 ^ 1 + cos * 2 1 



{n - n l ) n 



— {A X B X + A 2 B 2 ). 



On putting for 4i, &c, their values, and multiplying out, the latter 



factor becomes 



N(, r (C-B A-C\ „ IA-C C-B\ 



This may be much simplified for bodies nearly spherical, and might be 

 put into the form 



N l£ y + M x ( s C-AC-B 



I) 2 AB V" ni AB 



