213 



is augmented by 



% A X -B X . i 



£ sm 2n - n 1 : 



2 2n- n 1 



and the terms in 0 and ^ are augmented in the same manner. Adding, 

 then, these additional increments to those just found, we shall have for 

 the whole increments depending upon P, 



, = t + 1 (Al^) p cos (2» - n) I P cos 2^T3^ 



J 2rc - w 1 * 2rc - 



cos i . A x - Bn . ; cos * , A x - B 2 _ . - — — — . 



0 = nt = - — 1 — - 2 Psm2w-w l + — I — — — P sm 2w - 3w ! ) 



sm i J 2«-)i 1 ■ , sm t 2n - 3n ] 



+ P sin 2n - 2n l 



f = A 1 1 '-- Psm2n - 2ft* 4- ~ i f 1 "f 2 1 Psin2w-3^ 

 sm * ' 2» - n 1 sm < * 2rc - 3ft 1 



These terms must be added to those given in i, &c, by the first ap- 

 proximation, and then substitute for i, &c, in N. Also it must be re- 

 membered that corresponding to the term sin i sin 20 - 9, which arose 

 from the multiplication together of sin i sin 9 sin 20 - 20, there will 

 be another term, sint sin (20 - 30). Making the substitutions, there- 

 fore, we shall have 



(11 1 1 



sin i (sin 20 - 9 - sin 20 - 30) = + - 



v ^ ^ ' V2 2n - n x 4 2n - 3n l 



--\PA X -B 2 



n J 



which will therefore give the term in N 



15/1 1 1 1_ l_\ 



2 \2 2n - n 1 + 4 2n - 3n l " n 1 J 1 " 2 . 



the last part of this, however, namely, that multiplied by - — will 



I vanish ; for the term sin i sin 9 sin (29-29 1 ), will produce in term in 0 

 of the form Q A x - B. z cos 2n - n\ which, when introduced into the 

 first part of N, viz. 



B -A . 



sm 29 - 26" 



O 



H. I. A. PROC. VOL. X. 2 G 



