[ 46.5 J 



^XLIV. On the Potentials of Uniform and Heterogeneous 

 Elliptic Cylinders at an External Point. By N. R. Sen, 

 M.Sc* 



nPHE potential of an infinite elliptic cylinder at an external 

 -L point is generally expressed in the form of an integral, 

 and it is well known that a transformation in conjugate 

 functions would allow the same integral to be represented by 

 a much simpler expression*}"- It is here proposed to express 

 the potential in trigonometrical series. The method followed 

 is that of integration, which will be shown to be applicable 

 also in certain cases of heterogeneity. It will be found that 

 the potential is always expressible as 



at ^ a c °s nd 

 A logr — 2A» — , 



where A„ in its most general form can be expressed by 

 hypergeometric functions in e 2 (eccentricity), reducing in 

 two special cases to finite binomial forms. This happens 

 when the cylinder is homogeneous or when the density 

 (supposed constant along lines parallel, to the axis) at any 

 point on the elliptic section varies inversely as the focal 

 distance of the point. We shall simplify our problem by 

 considering only the logarithmic potential of the elliptic 

 section to which the (Newtonian) potential of the elliptic 

 cylinder is equivalent but for an infinite constant and the 

 constant multiplier 2. 



Before proceeding with the solution of the problem pro- 

 posed above, it would be useful to consider the expansion of 

 (1 + e cos <j>)~ 11 , e<l in cosines of multiples of <£. 



Expanding (1 + <? cos <f))~ n by the Binomial theorem and 

 replacing the powers of cos (/> bv cosines of multiples of <£, 

 it may be shown that J 



CO 



(l + ecos<£) - " = 2 A" cos m</>, 

 where 



m [ } (n-l)! m!*2—i' L V 2 ' 2 >*" + *■>*)>*<*>: 



where F is the hypergeometric function of the four elements 



* Communicated bv Prof. D. N. Mallik, Sc.D., F.R.S.E. 



f Lamb, Mess, of Math. 1878. 



X This expansion in another form is given by Gauss. 



Phil. Man, S. 6, Vol. 38. No. 226. Oct. 1919. 2 K 



