386 
MR. W. D. RIVER OR CERTAIR DEFIRITE INTEGRALS 
f 
J n 
= (-!)«-T 
v ' 1.3.5 
(4?i +1) 
A+-+? 2 A. (1 + s gM-, 2»A„ 1+ &c.). (14) 
4 n + 3 
(4jH-3)(4?H-5)1.2 
In certain cases this series admits of summation. For example, let us examine under 
what circumstances jj/A~' ; P 2 +S admits of being evaluated 
Hence 
kfk 
9 \ 9 
n\ 
>1+1 — 
+n 
n +1 
If we now write 
A«+o— 
|+ w )(§+ n + 1 
(%+l)(% + 2) 
« = 7 ) 
/3= n +1 
A„, &c. 
we find for the integral f j/r *P 2 , 2 c?S, the expression 
where 
/ i \ A(^ + 2)(A; + 4) . . . (k + 2n — 2) 
' ' (2n + l)(2n + 3)(4?t+l) 
^ x _|_^_p a ( a+1 ) 
7(7 +1)1.2 
r(7)r( 7 -/3-«) * 
T(7-/3)r(7-«) 
r(2n+-)r 
1 —k 
r(n+g]r(n+l + 
1 -k 
n+- . . . 2 ra+- 
l-fc 
1-& 
+ 71 
* Bertrand, ‘ Calcul. Integral,’ 1870, pp. 495-6. 
