THE 

 LONDON, EDINBURGH, and DUBLIN 



PHILOSOPHICAL MAGAZINE 



AND 



JOURNAL OF SCIENCE. 



[SIXTH SERIES.] 



AUGUST 1^0?>. 



XXIY. On a New Mode of Expressing Solutions of Laplace'' s 

 E^juation, in Terms of Operators involving Bessel Functions. 

 By L. N. G. FiLON, M.A., E.Sc, Lecturer in Mathematics 

 and Fellow of Unirersitij College, London^. 



1. TN a recent number of the R. A. S. ^ Monthly Notices'' 

 JL (vol. Ixii. no. 9, pp. 617-720), Mr. E. T. Whittaker 

 has shown that the general solution o£ Laplace^s equation 



d^> d^ ^i!Y_o 

 r/.i'2 -^djf "^ dz^ 



could he written in the form 



_/ (x cos r + ?/ sin L- + iz, v)dv, 



where / is an arbitrary function of the two arguments 

 X cos v+y sin v + iz and r and i — \/ — 1. 



It is further easy to show that the function/, which, by 

 Mr. Whittaker^s proof, must be an analytic function of the 

 first argument, need not be an analytic function of the second 

 argument. All that the proof assumes is that the function/, 

 treated as a function of the second argument, shall be ex- 

 pansible in a Fourier Series, the coefficients of which are 

 analytic functions of the first argument. This merely re- 

 quires that /, as a function of the second argument, shall 

 have only a finite number of turning-values or of finite 

 discontinuities, 



* Communicated by Prof. M. J. M. Hill, F.R.S. 

 PUL Mag. S. 6. Vol. 6. No. 32. Aug. 1903. 



