84 
i’KOFESSOK A. 8CHUSTEK OX SOME DEFINITE INTEGRALS, 
Equation (B) is the result of cr ditferentiations of (2), and multiplication by 
sin°"^^ Q. (C) may be proved by combining 
w 
ir (IV (FV 
n , ,2\ (• ±71 'f -L H » ±11 
- Z'-) 7 ,^ = (1 - 77TT. - - .T, .T - 1 --T 
itb tlie fundamental equation for zonal harmonics. The latter leads directly to 
<1 
(1 -f"') ,/.. = - 7 ;:-. A" + !)• P« 
dfjF 
from which we derive (by ecpiating the two expressions on the right-hand sides); 
^ fZ"P 
c// 
7<T+lp 7crp rr~^P 
/ 1 o\ « o ± n / . \ / I T \ ” 
= <" -r + 1)7^ 
or, after multiplication by (1 — /x-)-, 
sin = 2pcrQ,; — (?7 + cr)(>Z — cr -f l) sill 6Qr\ 
If p-Q." lie now substituted from (A) and sin 0 from (B), the equation (C) is 
obtained. 
If a — 1 be written for a in (B) and cr + 1 for cr in (C-) we may combine the two 
equations, so as to give (D) and (E). (E) is an important relation obtained from (B) 
and (C). The formulee (G) and (H) are easily derived liy direct ditferentiation and a 
few simple transformations. 
If the equation (H^) is integrated with respect to /r between the bmlts — 1 and 
“b 1, the left-hand side vanishes at both limits : hence, after changing from cr + I 
to cr, 
+ o- - 
1 ) (n 
+ j 
cr + 2) - djx, 
and, by applying the same process to the rigiit-band side. 
~ cr _ 4 ^ ~ 1 ) ^ ~ 3) ^ "I" — «■ + " c//x. 
Repetition of the same proceeding will ultimatelv lead for even values of cr to 
