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XXIV. On the Values of the Integral ^ Q n Q n , dp, Q n , Q n , being Laplace’s Coefficients 
of the Orders n, n', with an application to the Theory of Radiation. By the Hon. 
J. W. Strutt, Fellow of Trinity College, Cambridge. Communicated by "VV. Spottis- 
woode, F.R.S. 
Received May 17, — Read June 16, 1870. 
In the course of an investigation concerning the potential function which is subject to 
conditions at the surface of a sphere which vary discontinuously in passing from one 
hemisphere to the other, it became necessary to know the values of the integral 
s: 
Q.„, Q„, being Laplace’s coefficients of the orders n, n' respectively. The expression for 
Q„ in terms of p is 
1.3.5....(2»-l) f n(n- 1 ) », w(n-l)(n-2)(n-3) ,_ 4 
1.2.3 ..n V 2 . (2?i — ly 2 . 4(2n— l)(2n— 3) ^ 
Q.= 
i_ )* . 
but the multiplication of two such series together and subsequent integration with re- 
spect to gj would be very laborious even for moderate values of n and n 1 . 
By the following method the values of the integrals in question may be obtained with- 
out much trouble. According to the definition of the functions Q, 
; 1 T'Q 1 6d-Q 2 <? 2 4" • . • ~\-Cl n e n -\- . . . 
■y J-TC — 
so that 
V 1 + e 2 — 2qx 
(IjJ. 
V 1 + e 2 — 2 eft. V 1 + e n — 2e\ 
71 = 00 TV = 00 
■ V. V 
H' 71 = 0 7l' = 0 Jo 
QnQn'dp . e n e' n ', 
which shows that j Q„Q n ,dp is the coefficient of e n e' n ' in the expansion of the integral 
on the left in powers of e and e'. 
On effecting the integration and reducing, we obtain as the quantity to be expanded, 
1 , (l + V ee')( V e— V e') 
lo£ 
;=A5 lo s(i+y«0 
pee 1 & Ve V 1 + e 12 — V d V 1 + e~ V ee' 0 v 1 v J V ee' 
ed {ed)% (ee 1 ) 2 
log\/l-i -e 2 + log' 
j 
MDCCCLXX. 
* Thomson and Tait’s Natural Philosophy, p. 624. 
4 K 
