4 Mr. W. D. Niven. On certain Definite Integrals. [May 1, 



In the "Proceedings of the Royal Society," vol. xxvii, pp. 63 — 71, 

 Professor J. C. Adams has found the value of the integral 



P^P^Pf-dS. 



Mr. Ferrers had also independently succeeded in the evaluation, 

 and had published the result in his " Treatise on Spherical Harmonics." 

 In the paper of which this is an abstract, the same integral is dis- 

 cussed, and an attempt is made to extend the result to the cases of three 

 tesseral harmonics, and of four zonals with coincident poles, but with 

 only partial success. 



2. If we seek to apply the same methods to ellipsoids and- ellipses, 

 we find that 



if 



VprfS, 



taken over the surface of the ellipsoid of semi-axes a, b, c, where p is 

 the perpendicular from the centre upon dS, is equal to 



rP d 2 r? 2 V 



dv dz/ 



dx dy 



o 



and that 



/ 2 d 2 . „d l , a cPv* 

 \\\Ydx dy dz=^abc§- \ dx dy <hj_ y 



As a particular case, let us consider V as the potential due to unit 

 matter at the point (/, g, K) outside of the ellipsoid. Then, since 



and therefore 



VQs=(f-xy+(g- y y+(h-z)% 



d_ 1 __ d 1 

 dx PQ of PQ' 



■dx PQ df PQ' *' 



it follows that we may take x, y, z, zero before differentiation, and 

 write the two results : — 



PQ 



A df da dlj 



2t.+ l! Jp +g > + K 



(3) 



a relation giving the potential of an ordinary ellipsoidal shell 

 spherical harmonics ; 



