TRAJJSACTIONS OF SECTION A. 645 



when the angle of inclination is — (n, m positive integers). Illustrations : 



m 



": {m=l, -I); '^(m = l,n = 2: m = 2,w = 3). 

 m m 



So far as I can learn it was first pointed out by Sommerfeld,' that a Riemann's 



surface might be here employed, and with the help of a Multiple Space, problems 



in Electrostatics, Electric Conduction, Hydrodynamics, Sound, and Conduction of 



Heat, in which the boundaries are planes meeting at an angle ^, have now been 



m 

 solved by Images. 



The ideas involved may be illustrated by the potential problem in which the 

 boundary is the semi-infinite plane 6 = 0. 



There is nothing to prevent us assuming that in this case the space with which 

 we have to deal is defined by the range of o< 6< 2it, and the behaviour of the solu- 

 tion of the equation V"" = o, outside that rapge need not concern us. It is found * 

 that the periodic fimction 



u = 



■ E '•"- V^'' 



where R'^ = r^ + ,■"' - 2rr'cos {6 - 6') + (= - s')-', 



2rr' cosh a, = ;•'- + ;•"- + (s- s')'-^, 



a = cosh ' , 



and T = cos 



i^^y 



is a solution of the equation V '" = "i with the required properties, its only singu- 

 larity, in the range -27r<<9<^7r, being at (/•', ^', s'). CaUing this the solution 

 corresponding to a pole at (6'), if we associate with it that due to a pole at (-6'), 

 we obtain a function, satisfying the boundary conditions, with only the one singu- 

 larity in the range with which we are concerned. 



Similar ideas enter into the solution of the other questions of this nature : 

 where the ordinary Image method would reproduce singularities in the original 

 space, by taking a function of a suitable period (a Riemann's space of the proper 

 order) this result is avoided. 



The problems in potential involving spherical boundaries may be solved by 

 Inversion from the above. Hobson has considered,^ by discussion in series, the 

 direct solution of the cases of the circular disc and spherical bowl. By a method 

 similar to Sommerfeld's, I have found the solution for the general case. 



Taking the coordinate system employed by Hobson, in which the position of 

 the point P is defined by 



1 PA, 

 P = log p^3' 



e= ZAPB, 



(p = azimuthal angle, 



the boundary d = constant, represents a portion of a spherical surface, of which the 

 rim of the circular disc, whose diameter is AB, is the base. If we are dealing with 

 the space bounded by an infinite plane with a circular hole, we define this 

 region by 



o<d<2n, 

 and associate with it that defined by 



-2Tr<6<0. 

 Corresponding to the singularity at {6'), we shall have one at {-6'). 



' -Vath Ann. ■' Proc. Lond. Math. Soc, xxviii., p. 413 



^ Tram. Camb. Phil. Soc, vol. xviii. 



