388 
MR. W. D. NIVEN ON CERTAIN DEFINITE INTEGRALS 
First let n>m, then in S we Lave n —m positive and m negative factors, and in that 
case 8 becomes 
, . 2n—2m\2m\ 
(_ iy» - 
\ / 2 2n n — m\m\n\ 
Next, let n<m, then S becomes 
(-O' 
2 m\m—n\ 
2 2 “2m—2 n\m\n\ 
Substituting these values of S in the expression for the integral we find 
(1) if n>m 
( —1) m 27t 3 R 2 2to! 2 n\ 2n\ 2n — 2ni\ 
2&B+4 n n s r m\ n—ml m\ m\ n\ n\ 
■ ■ ■ ■ (16) 
(2) If n<m 
( — l)"27r 2 R 3 2m! 2m! 2 n\ m—n\ 
. . . . (17) 
22m+4?i m _j_ n j 2m — 2n\ m\ m! n\ n\ 
I 
Q,Sys, 
d% 
§ 5. The results for the two integrals now stated are, if a be the colatitude of the 
pole of Qi, 
47rRfP,(cos a).. (18) 
2tt'R 2 0' —^)0'+id-l)P,(cos a).( i9 ) 
where, in the second integral, j—i must be a positive even integer. 
GENERAL THEOREMS IN DIFFERENTIATION. 
§ 6. Before we proceed with the remaining cases, it will be convenient to state and 
prove various theorems in differentiation.'" 
Theorem i. The general operator (7) upon the reciprocal of r may be made to 
operate, instead, upon a homogeneous function in x, y, z of the i th degree; this 
effect being expressed by the relation 
.. . din d\dl h . . . .( 2 °) 
where the pole of Qi is in the direction of r. 
* Theorems i. and ii. were given in a paper bj the author in the ‘ Messenger of Mathematics,’ No. 73, 
1877. On account of the brevity of the proofs, and in order to secure completeness, it is thought best to 
reproduce them here. 
