384 
MR. W. D. NIVEN ON CERTAIN DEFINITE INTEGRALS 
It is obvious the same method of solution as the foregoing will apply generally 
towards determining 
where f(p) is any function whose differential coefficients are finite at the origin. In 
fact, we find 
j]7( Z )P,4S=7r(2R)*y 
ni 
R 2 
d n+2 
Pd 
d“ +i 
n + l\[dz ' 2(2n+3ydz 1 2.4(2?H-3)(2w + 5) dz 
+ &C. f(z). (12) 
where, after the differentiations are performed, z is to be put equal to zero. 
A result practically equivalent to the above is given by Heine (“ Kugelfunctionen,” 
second edition, p. 76), and applied to various cases where the functions can be ex¬ 
panded in simple series of zonal harmonics. The functions are 
(l+Jcn 2 ) fo+l) , (1— /A)*, sin V, (1— /A) “ 
Similar functions are also to be found expanded in Todtiunter’s “Functions of Laplace, 
Lame, and Bessel,” §§ 48, 145-147. We shall now discuss more general cases of some 
of these functions. 
§ 3. The case now indicated will be discussed only for integral values of m and n. 
Let us consider the integral 
jl ^+y 3 )* ^, ^ jj^ +yS y ud $ 
If we use the suffix 1 in any operator when it is upon (a; 3 -}- y 2 ) m , and 2 when it is 
upon u, we have as the general term operating, 
4ttPA +2 // d d V.( d d y V 
2i + l!\\efe 1 dxj \dy 1 dyj dzj 
or, since 
d d ddd d 
dx 1 d.Xq d dr) 1 d£ 2 dy 2 
d d _ . ( d d d * d \ 
dy x dy~^ \d %i d Vl d% 2 dyj 
the general term above is 
47rPA +3 / / d d y d d \ d 2 V 
'2i + lTV 4 Wi dyj ' dz 2 ) 
