WITH APPLICATION TO SPHTHilCAL HARMONIC ANALYSIS. 
199 
terms, owing to one of the factors in the product (n — i) {n — p — 1) . . . 
becoming zero. In the special case n = cr the value of reduces to 
(2n- - 1) !! (n + 1)! _ 
(n — p + 1 )!! {v + p 4- 1 ) I! 
The above equations are inconvenient for numerical calculations, unless n — cr be 
small, or — p be even and small. We obtain quite different and generally more 
convenient expressions, if we begin by expressing sin pd or cos p6 in terms of a series 
proceeding by powers of sin 6. 
Let 
T3_1.-R .p —P~ . p _ p - 2 .p .p+ 2 
Lq — 1 5 Lj — , Ho — ^2 > L, — 12 3 
and generally 
=: 
(p — A + 1) (p — A 4 * 3) . . . (p + A — 3) (p + A — 1) 
A 1 
Also put 
_ (p 4- A 
“ (p - A - 1)! ! A ! ■ 
C„=l; G, = p.-, C, = p.N C,=p. 
p — 1 . 7? 4- 1 
; = P . 
p — 2 . 7 ) . p 4- 2 
= V 
1.2.3 ’ ■ 4; 
(77 — A 4- 2) ( 7 J — A + 4) . . . (p + A — 4) (t^ 4- A — 2) 
A ! 
p (p + A-2)!! 
A! (p - A);; 
Then the well-known expressions for the trigonometrical functions of the multiples 
of an angle may be written : 
If j) be even : 
sin p^I 
cos 6 
= sin 6 — Bo sin^ 6 -f B- sirr' 6 
cos pO — Oq — Ch sill' 6 + sin'*' 0 
and if p be odd : 
sin p6 = C| sin 6 — sin^ 0 C- sin'"* 0 
cos p9 ^ d + Bi siiB 0 
cos O’- 'I 
We derive from this, p> being even : 
± B^_i sliH * 0 , 
± Cy siiH 0 ; 
dz sini" 0, 
zh B^_i sin^~* 0 . 
+1 
-1 
+ 1 
( 
-1 
A = /.< — I 
A-1 
sin p0 c/p — 2*-* (— 1) ' 1\ sin'^ 0 cos 0 dp 
A=1 
\=p 
+1 
-1 
Q;; cos p0 dp = S'-’ (—1) Cm siiH 0 dp, 
A = 0 
+ 1 
