Dr. J. H. Jellett on Laplace's Equation. 401 



and the equation becomes 



(n+l)v = 2ba; (2) 



or, as it may be otherwise written, 



(»+l>=2&J, (3) 



where r is the distance from the centre of the sphere of a 

 point situated on the radius through the point of contact. 



To prove this, Laplace assumes in the first place that the 

 thin shell lying between the sphere and the original surface 

 may be replaced by a " distribution" of matter (to use Gauss's 

 expression) on the spherical surface itself. This assumption 

 is only approximately true, inasmuch as it neglects the differ- 

 ences of distance among the particles situated on the same 

 radius vector. The degree of the approximation will be con- 

 sidered further on. 



Let P be the point of contact, and Q any other point on 

 the surface of the sphere. Taking the origin, 0, at the centre 

 of the sphere, put P Q = #, PQ = p, and let e be the density 

 of the distribution at the point Q. Then evidently 



a = b 2 \ 'f^cp" cos QPO sin 6 d6 d<f>. 



*^o Jo 



Jo Jo 



{n-\-Y)v=-fr\ I ep n + l &\\i6ddd<$>, . . . (4) 



But 



whence 



20PcosQPO = 2PQ = /3, 



H27T 

 6p n +* sin 6 dddcf). ... (5) 

 j 



Hence (n + l)v — 2ba; and Laplace's equation is proved, if the 

 foregoing integration be legitimate. It is of course only ap- 

 proximately true, the degree of approximation being the same 

 as that of the assumption made in the beginning, namely that 

 a distribution on the spherical surface may be substituted for 

 the thin shell. 



We have then to inquire, in the first place, what conditions 

 are necessary to the validity of the integration indicated in (4) 

 and (5), and, secondly, to what degree of approximation is the 

 result true. 



In reply to the first question, it may be said that it is suffi- 

 cient to ensure the validity of the process of integration that 

 the function to be integrated remain finite within the limits 

 of integration. Thus, if the integral be \\ U d6 d<f>, it is 



