335-337] Multiple-valued Potentials 281 



we see that the infinities of f(u) occur when u = a, a + 4?r, a + 8?r, etc., and 

 the residue at each is unity. Hence, if we take the integral round one 

 infinity only, say u = a, the value of 



will become identical with -^ at the point at which R' 0. Moreover, 



expression (263) is, as we have seen, a solution of Laplace's equation : it 

 is seen on inspection to be a single-valued function of position on the 

 Riemann's surface, and to be periodic in with period 4?r. Hence it is the 

 potential-function of which we are in search. Thus 



du 



The details of the integration can be found in Sommerfeld's paper. The 

 value of the integral is found to be 



where r = cos J (0 a), cr = cos (p p). 



337. Other systems of coordinates can be treated in the same way, and 

 the construction of other Riemann's spaces can be made to give the solutions 

 of other problems. The details of these will be found in the papers to which 

 reference has already been made. 



REFERENCES. 



On the Theory of Images and Inversion : 



MAXWELL. Electricity and Magnetism. Chap. xi. 



THOMSON AND TAIT. Natural Philosophy. Vol. n. 510 et seq. 



THOMSON, Sir W. (Lord KELVIN). Papers on Electrostatics and Magnetism. 



On the Mathematical Theory of Spherical and Zonal Harmonics : 

 FERRERS. Spherical Harmonics. (Macmillan & Co., 1877.) 

 TODHUNTER. The Functions of Laplace, Lame', and Bessel. (Macmillan & Co., 



1875.) 



HEINE. Theorie der Kugelfunctionen. (Berlin, Reimer, 1878.) 

 MAXWELL. Electricity and Magnetism. Chap. ix. 

 THOMSON AND TAIT. Natural Philosophy. Chap. i. Appendix B. 

 BYERLY. Fourier's Series and Spherical Harmonics. (Ginn & Co., Boston, 1893.) 



On confocal coordinates, and ellipsoidal and spheroidal harmonics : 

 TODHUNTER. The Functions of Laplace, Lame, and Bessel. 

 MAXWELL. Electricity and Magnetism. Chap. x. 

 LAMB. Hydrodynamics. Chap. v. 

 BYERLY. Fourier's Series and Spherical Harmonics. 



