324 Mr MURPHYs THIRD MEMOIR ON THE 



By Art. (7), 



2-4 

 Q„ = coefficient of Jf in -^ {^ + 1 - A (1-2^)}^, 



in which R represents |1 — 2// (1 - 2/) + /i'}*. 



Putting for t its value in terms of 0, we obtain 



J? = {1-2/i cos^ + /i^}-i = (l-Ae*^^)i.(l-//e-'''^^)^^ 

 and l-/i(l-2it) = l-/4cos0 = 1(1 -Ae*^^) + ^ (1 -//e"*^^). 



Hence, ^ + 1 -/i (1 - 2/f) = |{(l-^e*^^)^ + (1 -//e-*^^)-^''; 



therefore, Q„ = coefficient of /r in x . Jj " ^t-v-!!-^4' "r'^^! 



= ^ coefficient of A" in (l-/<e*^^)-* + (1 — Ae-*^'^)-* 



= c 



c . cos n<p] 



13 5 (2« — 1) 

 where c = ' ' '"^ ^ , the limits of ^ are and w. 



2.4.6... 2ra 



II. To evaluate q„ when y« = — i. 



As above, we have (p = eos"' (1 — 2t), 



and q„ = coefficient of h" in ^-^ . {^- 1 +/< (1 — 2/)}''. 



But i? - 1 + /. (1 - 20 = i p-^;!:l'^>' - (i::.^-_:!:^^)H'^ . 



I V -1 V-l j 



-. q„ = ^ coefficient of /«" + ' in 



\/-i V-i 



c — . = c sin (1 + n) (p, 



* v — 1 



, 1.3.5...(2m + 1) 



^^^••^ ^' = 2.4.6■.■(2;^4-2) ' 



