190 
PROFESSOR A. SCHUSTER ON SOME DEFINITE INTEGRALS, 
which gives 
|"’Qr=Qr</,. = 4 (<7 + 1). 
When n = i, that part of the integral which depends on P.-^i 
that case 
3 _ {n + a) ! 
'In +1 {ii — a — 
vanishes, so that in 
More particularly when cr = 0 we have the integrals, 
•’-1 
= 4, if n + % is even and i > n 
= 0 in all other cases, 
— o ~ ^ 
“ ~ + 1 ■ 
r+i 
The result that Q.n^'^QUdjx has zero value whenever i '> n can be extended to 
r+l 
the more general integral j provided then cr > p. To show this we need 
onlv consider the series, 
Q: = AoQ: 
AaQ^+Q + • • • 
where the coefficients may be determined from the integrals found above. Multiplying 
both sides with Qf”'* and integrating, all the terms vanish when i > n and hence 
j Qn Qr ^ = 0 in that case. 
From cr — 4 we may proceed to o- — 6 and so on. 
QlQldp, we use formula (F), multiplying both sides by 
•' -1 " _ 
and integrating. We may without loss of generality take p to be smaller than cr. 
Utilising the result which has just been obtained, one integral on each side drops 
out and the equation becomes for even values of cr 
+■1 
-1 
= - (n - a- + 1) (.» _ cr -b 2) 
= (?? — cr + 1) (n ~ cr + 2) {n — cr -b 3) (n — cr + 4) | Ql~*Qndp 
= {- 
-P {n -p)] r+l 
(« — cr) 
■ •'-1 
(-1)-^ 
(n + p) ! 
{71 — cr) • '2/1 + 1 
if p < cr and p -b cr even 
= 0 if p + cr is odd. 
