472 Mr. N. R. Sen on Potentials of Uniform and 



find this would be a decreasing monotonous sequence tending 

 towards the limit zero if we take 



r>2c, 



> SS*, 



where 8' is the second focus. We can also show by- 

 applying the usual ratio-test that the series is absolutely 

 convergent under the same conditions. This shows that in 

 addition to the outside region the series is also convergent 

 inside the area lying betw r een the previous limiting circle 

 (drawn for the purpose of integration) and a concentric 

 circle whose radius is SS'. Consequently the present form 

 of the potential function is valid at all points inside these 

 two circles (and outside the elliptic area as we are dealing 1 

 only with the external potential). 



It would seem that we are incapable of accepting the 

 potential function in the present form of the infinite series 

 inside the circle of radius SS'. But in fact the region in 

 which this trigonometrical series fails is much more limited. 

 If we take S' as our origin and proceed to find the potential by 

 the present method we get the same series, which in a similar 

 manner may r be shown to be applicable everywhere outside 

 a circle radius S'S. In general, these two circles bounding 

 the regions of convergence overlap outside the elliptic area, 

 and it is only inside the two small areas common to the two 

 circles and symmetrical about the minor axis that the present 

 trigonometrical series fails. Excepting this common portion 

 the present form of V would hold good everywhere, only we 

 should take care to choose the origin properly— measuring r 

 from S or S' according as the point lies inside the circle of 

 centre S' or S. 



It is curious to note that the convergence of the series 

 depends on the eccentricity of the ellipse. The two limiting 

 circles would have their common portion entirely within the 

 elliptic area if 



SS' ^SB where B is an extremity of the minor axis, 



i. e. 2ae < a, 



i. e. e< \. 



This shows that when the eccentricity of the ellipse is not 

 greater than -J- the function V gives the potential everywhere 

 outside ihe elliptic area, with judicious choice of origin. 

 This includes the important case when the ellipticity is 

 small and the ellipse is obtained from a circle by a slight 



