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LIII. On a Peculiarity of the Normal Component of the 

 Attraction due to certain Surface Distributions. By 

 Ganesh Prasad, M.A., D.Sc, Professor of Mathematics 

 and Principal in the Hindu University of Benares *. 



r|lHE object of this paper is to point out certain surface 

 JL distributions for each of which the component N of 

 the Newtonian attraction at a point P along the norma), 

 which passes through P and meets the surface at a point 0, 

 tends to no limit as P approaches along the normal. 

 It is believed that such surface distributions have not been 

 pointed out by any previous writer. 



1. At P, let N be equal to N! + N 2 , where N x corresponds 

 to a small area S round 0, and N 2 to the remaining part of 

 the surface. Then it is obvious that the limit of N 2 is 

 existent ; we have to consider the limit of N^ For the 

 sake of simplicity, the surface may be taken to be regular 

 in the neighbourhood of O and, consequently, S may be 

 taken to be a circle of centre 0, radius a, and density a. 



Case 1. a— cos loo; —■'. 



2. First let cr = cos log -, where r is the distance between 



and the point Q where the density is a. Then it will be 

 shown that the limit of N x is non-existent. 



Divide the circle S into thin concentric rings. Then, 

 taking the origin at and the axis of z as the normal at 0, 

 we have 





Thus we have to investigate 



LimNj, i.e., -2wUm f* 



"•I 



2\3/2> 



*=o ,=oj o (1+**) 



where zt = r. 



3. Now let C be a sufficiently large quantity independent 

 of 2. Then 



r a ' z tadt _f c P*/* 



X (uW 2 "Jo + Jo 



tadt 



(l + t 2 f 2 ' 

 * Communicated by the Author. 



