Heterogeneous Elliptic Cylinders at an External Point. 4G9 



would have their coefficients in finite forms which it is easy 

 to calculate. 



A very interesting case of the above arises when p=—l 

 or the heterogeneity is of such a nature that the density at 

 any point on the elliptic section varies inversely as the focal 

 distance. Since 



A;:+1 = (~ 1)n Jrfin -2^- *(»+£' "+ 1 ' n+1 ; e2) 

 (2n)i ^_ • i 



v } n\ n\'2 n -S'{X-e?)«+V 

 we have in such 



a case 



V 



= A ' log r- | -f±~ . ( -) cos n&, 

 ** ?i(n+ 1) \rj 



»=i 



and miking the above substitutions the potential function is 

 found to be given by 



<*=*£ V = 2Iog,-S(-l).- , ( 2 ,"1 ! 1w f C )'cosnfl. 

 tt/ ° ■ n=1 v ' 2f-. 1 n.w! (n4-l)!V^/ 



(ii.) Suppose that the density <r=/o p sin <?(£. 

 Then as before 



i * sin qdtdd) , 



loor 



(P + 2)J_ ff (l-H?cos0)P+ 2 



:(n+p + 2)r' l J- 7r (l + e~cos<l>) n+ P +:t 



-Hr n(n + p + 2)r' l J_ ir (I 



Now 



sin qcf)d(p _ n 



.,,(! + « COS <£)*+*" 



i 



and replacing the product in the numerator of the other 

 integral by the sum of two sines we have 



vr oo \n+p+2 in+p+2 /J.n 





sin nO. 



The logarithmic term is absent. The line = 0, = it 

 is a line of zero potential, as is obvious. Also at a great 



distance from the origin where ,, -., etc. can be neglected 



r~ r , 



