101, 102] NEWTONIAN POTENTIAL FUNCTION. 193 



by Poisson, and finally in a rigorous manner by Dirichlet. A 

 proof due to Darboux is given by Jordan, Traite d' Analyse, 

 Tom. II. p. 249 (2me ed.). 



102. Potential of Circular Disc at points not on axis. 



We have found in 81 the potential of a disc of surface 

 density <r, radius R, at a point situated at a distance r from the 

 center on the axis to be 

 (i) 7 



Developing by the binomial theorem for the two cases r < R, 

 r>R, 



(2) 7= 



1 r" 1 1 r 4 1.1. 3 r 6 ) 



<> r<R 



F-amrtT-ix-r.-3-j -,j 



IE 2 LIE 4 1.1.3-R 6 



FIG. 46. 

 If now the point be not on the axis, but on a line through 



the center making an angle 6 < =- with the axis, and at a distance 

 r from the center, we may put 



(4) V= 27TO- E - rPj (cos 0) + P 2 (cos 0) 



(5) 



For both of these series are convergent under the assumptions 

 made, both are harmonic, for r n P n and P n /r n+1 are zonal harmonics, 

 and both take the given values when = 0. 



W. E. 13 



