On Certain Problems of Two-Dimensional Physics. 761 



electron or even a positive ion (of smaller radius so as to 

 possess greater mass) could rotate and something like a 

 magneton would be the result, even i£ the elastic constants 

 were not supposed to be infinite. Actually, however, the 

 electromagnetic " potential " energy will produce effects 

 analogous to those due to a mass density varying from the 

 centre to the circumference. By supposing \ and /i to 

 be infinite, the " semirigid " rotating electron (electron- 

 magneton) could still be used as an hypothesis consistent 

 with the Principle of Relativity. 



LXXXII. On the Solution of Certain Problems of Two- 

 Dimensional Physics. By J. R. Wilton, M.A., D.Sc, 



Assistant Lecturer in Mathematics at the University of 

 Sheffield"". 



1. A GENERAL method of solution of certain types of 

 J7JL physical problem, in which the boundary considered 

 consists of a single analytical curve, may be founded on the 

 obvious remark f that the transformation 

 x+t,y = X(T) + iY(j), 

 in which t = tj — i^ and X and Y are real when r is real, 

 makes the real axis in the t plane correspond to the curve 



*=Xfo), y=Y( v ), (1) 



in the x + iy plane. We may therefore take the equation 

 of any analytical boundary in the form 



?=0, (la) 



or, if Q = , n + i1;, we have 



= T. 



For the sake of brevity, we shall denote X(t?) by X, X(t) 

 by Xj, and X(#) by X 2 , with a similar notation in the case 

 oE Y. 



In the simplest type of problem we are required to deter- 

 mine a function ty from 



together with the conditions ^ = /'(^), ^~=F(tj) on the 



boundary, where dn is an element of the outward drawn 

 normal. The solution is 



* Communicated by the Author. 



f Cf. Forsyth, < Theory of Functions/ § 265, p. 624 (2nd edition) ; also 

 Jeans, ' Electricity,' p. 264. 



Phil Mag. S. 6. Vol. 30. No. 180. Dec. 1915. 3 D 



