Heterogeneous Elliptic Cylinders at an External Point. 473 



deformation. For the area within the two circular strips in 

 which the trigonometrical series fails it is not possible to get 

 by the present method a simple value for the potential 

 function V. Starting from the beginning, we have to divide 

 the elliptic area into two areas by a circle passing through 

 the point where the potential is sought such that every part 

 of the one area is nearer to the origin than the point while: 

 every part of the second area is further from it. We can 

 use two logarithmic expansions in the two areas and find 

 the potential of the two areas separately. The method 

 of procedure is the same as in § 9. We get the potential 

 both in direct and inverse powers of r. But as the expression 

 is not a simple one we do not propose to give it here. 



7. 



A similar investigation is possible in the case of variable 

 density. When the heterogeneity is of the nature we hate 

 assumed in § 4, we can show that at least outside the same two 

 strips of areas between the two limiting circles the series V in 

 § 4 is convergent. It should be noticed that a transler of 

 origin to the other focus in the case of heterogeneity would 

 entail a change in the law of density. But if we take density 

 to be a rational, algebraic, integral function of the coordinates 

 of a point, a transfer of origin would involve a change of 

 density of such a nature that the new distribution would still 

 be represented by terms of the form p p cos qcf) and p p sin qcj). 

 So these two cases are sufficient for our purpose. 



As before, applying Dirichlet's test to V in § 4 we get the 

 condition of convergence by making the coefficient of cos nd 

 steadily tend to zero. This leads to such a condition as the 

 following : 



Lt (2n+p + q + l)l ++* F ( n + 1+ P±3 



n4 + P+ / / ,n + , J + l;e^-,0. 



If p^.q every term of F is less than the corresponding 



1 

 term in the expansion of — 3— p+q ; 



so that F < 



(l-e 2 ) n+ ~2 + 2 



