Heterogeneous Elliptic Cylinders at an External Point. 471 



. So we are to find 'p and q such that any of the four 

 following sets of equations should be consistent : 



p = q j p = q "I p+l = q ] p+l = q-]^ 



<2 being zero or an integer. 



From these four equations we get only two possible 

 solutions, namelv, 



p = 01 p = -l 



? = or 9 = 



answering to the case of homogeneity and to that of the 

 density varying as the inverse focal distance. These are 

 the only two cases in which the potential function for an 

 infinite elliptic cylinder for the outside space is expressible 

 in a trigonometrical series with binomial coefficients. 



*6. 



In § 3 we have obtained a trigonometrical series for 

 the potential function V for the outside space by integrating 

 log r throughout the entire area of the section. But in 

 course of our analysis, in order to make the expansion 

 of the logarithm of the distance PA possible, we had to 

 introduce a certain limitation, namely, that r should always 

 be greater than p ; this immediately marks out a circular 

 area with centre S and radius equal to the maximum radius 

 vector within which the point A must not lie. It will now 

 be shown that the series V has a much wider area of 

 convergence which extends even into the limiting circle, 

 and consequently from considerations of continuity it 

 represents the potential function everywhere inside that 

 extended area. 



It is well known that the series X( — l) n a n cos n0 is 

 convergent if a ; - — >0 steadily. 



Considering the present series as a series of the same type, 

 we have 



2 (2/i+l)! 



a n = - 



,(!)" 



n n ! (n + 2) 



2 1^3 . 5 ... . (2n+l) . 2" . n ! / cY l 



/i(n + 2) ' n ! (n + 1)! \2r) 



2* 1.3.5 .... (2n + l) /So-V' 

 n{n + 2) '2.4.6 (2n + 2) \ r / ' 



