W'lTH APPLICATION TO SPHEKICAL HAPMONIC ANALYSES. 
L83 
[2n + 1) — ('H + 1) P/;+i + )^P,v_i 
(/P,_i rtT. 
(272 -f l) Pf, — 
"-1 
d/.i 
du 
From the.se e^juations we derive the following : 
[2n -F 1) IJ-Qn = {ll — cr +1) Qli + l + (^^ + O') Qn-l 
(272+l)sini9Q,l=:Q:::}-Q^!}. 
( 1 ). 
(2). 
(A) , 
(B) , 
= (/t + 0 -) (/2 + or — 1 ) Q;il} — (72 — (T+ 2)(72 — CT+l) . ( 0 ), 
O'' + O' —• 1) Qliil + Qpl.(D), 
= QUl + {>l ■— o -P 2) [v — rx + 1) Qn\ .(E), 
Qu ~ Qrt-2 = ('^ + < 7 " - 2 ) (72 + cr ~ 3 ) Qa-2~- (72 — 0-+ 2 ) (77 — cr + 1 ) . (F), 
(272 + I) y- shP 0 y 
^ ' dfji df^-^ 
-2/5 ' ^ "-1 
= (72 + cr) ( 77 +cr — /3+l)siu'‘ '+ ^ — (22 — cr+1 )(72“cr + /3) silp '0 
p-2^" ^ « + l 
d, 
silP 0 
d’^V 
( f L 
d/j.^- 
{2(x — p) silPd 
(T+ 1 
dp 
p{ii-p(j){n — cr+l)si!p ~0 
dp 
<r— 1 
(Cf), 
(H). 
As special cases of (G) and (H) Ave may put p equal successively to ex, cr + 1, and 
cr + 2. The following e(piations are thus obtained : 
(2/7 + 1) sin' 0 (p = (n + 1) (/2 + cr) — n [ii + I — cr) • (^0)’ 
2 sin 11 = GG' — (72 + cr) (72 — (x + I) Qr' .... (Hj), 
( 2/2 + 1) sin 0 '1^ sin 0 Q,! n {n + cr) GGi — ('^ + l) (" + I — cr) QGi • (GP, 
2cx sin 0m = {cr — 1) Q,F‘ — (cr + 1) (72 + cr) (72 — cx + !) . (H.), 
(2/2 + 1) sin~ 0 G;; = {n ~ 1) (77 + cr) Gl-i ~ (22 + 2) {11 — cr pi) GGi (Grs), 
2x siir ^ G« = (cr — 2) sin 0 GG^ — (cr + 2) (72 + a) (72 — cx + 1) sin 0 Gr' (Hg). 
The formula (A) is well known and may be obtained by cx ditferentiations of (1), 
substituting in tlie result an equation derived from x — 1 differentiations of (2). 
