124 



SCIENCE. 



[N. S. Vol. XVI. No. 395. 



{x^, y^, z-^) within the bovinclary is ex- 

 pressible in terms of its surface values and 

 the normal derivative of G. Thus the 

 solution of Dirichlet's problem is reduced 

 to a problem in integration when Green's 

 function is Imown. 



Some i-ecent advances have been made 

 in determining Green's function for cer- 

 tain boiindaries. To make them clearer 

 I shall begin with the simple problem of 

 finding Green's function for a region 

 "bounded by two planes at right angles and 

 extending to infinity. Here Lord Kel- 

 vin's method of images is directly appli- 

 cable. Let Po be the imag'e of the pole 

 Pj taken with regard to the first plane. 

 Let P3 be the image of P^ with regard to 

 the second plane; and P^ the image of P3 

 as to the first plane. Then the image of 

 Pi as to the second plane brings us back 

 to the first point, Pj. These' four poles 

 form a closed system, and there is only one 

 pole in the given region. The required 

 Green's function is 



in terms of the distance of the current point 

 {x,y,s) from the four poles; for this func- 

 tion, being a potential function, satisfies 

 Laplace's equation; it also vanishes on the 

 Ijounding planes by symmetry, and at in- 

 finity; moreover it becomes infinite as l/'r\ 

 at the pole P-^, and is infinite nowhere else 

 within the bounded region. 



It may be observed that a direct physical 

 interpretation of Green's function is illus- 

 trated by this problem. It is evidently the 

 combined potential due to a positive unit 

 of electricity placed at P^ and to the in- 

 duced charge on the bounding planes made 

 conducting and maintained at zero poten- 

 tial ; for this distribution realizes the 

 boundary conditions. Hence the in- 

 duced charge due to Pj is equivalent in 

 effect to three-point charges, namely, a 



positive unit at P3, and negative units at 

 P2 and P^. 



Next consider the problem in which the 

 angle of the planes is not an aliquot part 

 of -. The simplest case is when this angle 

 is 2-/3. Performing the successive re- 

 flections as before, it is found that there 

 are five refiections before the image comes 

 back to P-^. There are then six poles, of 

 which two are situated in the given region. 

 The function 



satisfies all the conditions except that of 

 having only one pole within the region. 

 It is thus not the required Green's func- 

 tion; and Lord Kelvin's method of images 

 does not furnish a solution. 



This method fails in two large classes of 

 problems: (1) When the successive images 

 (or poles) do not form a closed system; (2) 

 when more than one of these poles lie 

 within the assigned region. 



By the conception of a Riemann space, 

 Dr. Sommerfield* has recently made the 

 important advance of overcoming the diffi- 

 culty arising from the presence of two poles 

 within the region. He regards the whole 

 region as undergoing successive reflection; 

 and thus, in the problem last mentioned, 

 the whole of space is filled twice over. He 

 imagines a two-fold Riemann space having 

 the intersection of the planes as a winding 

 line, and one of the planes as a branch mem- 

 brane. The appropriate coordinates are 

 cylindrical (r, ff, z). The axis of z is the 

 line of intersection, and the plane z=o is 

 the plane passed through the original pole 

 Pi, perpendicular to the axis of z. The 

 radius-vector r is the distance of the cur- 

 rent point from the g-axis, and is the 

 angle which r makes with one of the planes, 

 taken as initial plane. 



* Proc. Land. Math. Soc, 1897, ' Ueber ver- 

 zweigte Potentiale im Eaiini.' 



