188 
ri’tOFESSOR A. SCHUSTER ON SOME DEFINITE INTEGRALS, 
(/i - 1):: A :;(/t - X - 2) 
^'/U:(-A- 2)!:(«4-A + 1)!!’ 
if n be even, other¬ 
wise the inteu'ral is zero. 
O 
Tills inchules, as particular cases, 
P„ siiF 6 dji = 0 if X lie even and ?/ > X + 1, 
•’ -1 
. (li - 1)A!'x:!(?i - A - 2);: . 
( — 1 ) - TT-’ 
;! (n + A + 1) ; 1 
{n - 1)!!A!: A!! 
n ; 1 (A + n + 1) ;! (A - ?t) !; 
f X be odd and n > X -h 1. 
If 12 < X + 1- 
Dr. W. D. Nivex (‘Phil. Trans.,’ vol. 170 (1879 I.), p. 379) has already obtained 
r+i 
an expression for tlie integral P„ slip 6 du. His results are Identical with the 
• -1 
above, when allowing for the difference In the notation and after correcting an obvious 
misprint in tbe equation marked (IG) on p. 388. 
If we write down the differential coefficients of P^, by means of the equation 
dV,Jdix = (2n - 1) P,,_, + {2n - 5) P„_,, + . . ., 
and repeat the same process p times, we olitain a series beginning with 
rPlW == 1- 3) . . . {2n -2^+1) P.-p + • • • 
There being no term containing a zonal harmonic of higher degree than 7i — p, we 
conclude that tbe above integral vanishes when e > ?i — p, and that when e = n — p 
f''p ^ 2 (2a--!)!! 
' V ^ ( 3 « - 2p + l)l! 
If e < 22 “■ p we may transform the integral as follows : 
L, IP A = 57i .L, 
1 r+Df(g2-iy r/pp„ 
Pm 
(//LiP 
dp. 
After e partial integrations, in which the first term always vanishes at both limits, 
the integral becomes 
f —IV 1 A+i 
