[ 138 j 



XII. On the Failure of Poisson' s Equation for certain Volume- 

 Distributions. By Gakesh Prasad. M.A., D.Sc, Sir 

 Rashbehary Glwse Professor of Applied Mathematics in 

 the University of Calcutta *. 



THE object of this paper is (1) to point out some typical 

 volume distributions for which Poisson's equation is 

 invalid, and (2) to prove that Professor Petrini's gene- 

 ralization f of Poisson's equation does not hold for every 

 one of these distributions. It is believed that the limited 

 scope of the validity of Professor Petrini's generalization 

 has not been pointed ont by any previous writer. 



1. Let p denote the density of the solid at any point 

 P (a, y, z) inside it. Then it should be noted that Poisson's 

 equation fails at P when V 2 V is either meaningless or has a 

 value different from — 4:7rp , Y being the potential due to 

 a small sphere of radius a and centre P. 



Some typical cases in which Poisson's equation fails. 



2. Case 14 Let the density of the sphere at any point 

 Q (£> Vi f) inside it be 



cos 2 <9 



'=— T' 

 log- 



where 6 is the angle made by PQ with the axis of s, and 

 r denotes the distance between P and Q. Then, remembering 

 that the external and internal potentials due to a spherical 

 shell of radius t and surface density P„(cos 6) are 



1^(1V +I and Mt(tT 

 2n+l \r J 2n+l \t f 



respectively, we have the potential at Q given b} r 



J: 



,4nrt t , 8tt<P 2 /t\ z l dt 



i-3--7 + -T5--(r)}— 1 



log, 



87TiP, / r\ 2 l dt 



•tt) 



, 1 



log J 



* Communicated by tke Author. 



-j- H. Petrini, " Les derivees premieres et secondes du potentiel," 

 Acta Mathematica, t. xxxi. p. 182 (1908). 



X 'This case has been studied by Prof. Petrini (loc. cit. p. 136), but the 

 treatment given by me is different from his. 



