196 Dr. L. N. G. Filon on a New Mode of 



So that 



If we suppose the constant 27r(—/)"* absorbed inside the 

 arbitrary function fm[z), the solution of Laplace''s Equation 

 may be written 



V = Jo (^/o ^ j /o 1-) + . . . + cos ?n(^ J„, (p j-Jj fm (^) + . . . 



+ sinm(^J.(^p .^^F..(.') + (8) 



the /'s and F^s being arbitrary functions. 



4. By giving different forms to the functions f and F we 

 obtain series of the various typical solutions of Laplace^s 

 equation. 



Thus, if we write 



/;U-)=Aefe (9) 



we get immediately the typical product form 



A^^^^]m<j>3,„{kp)e^^ (JO) 



5. If, on the other hand_, we assume 



Mz) = z-, ....... (11) 



then 



~rrM) - (m + r)\r\ ^^^^ 



If n is fractional the series actually extends to infinity, and 

 it is absolutely and uniformly convergent provided 



I h' < I -- 1 • 



If n is integral the series terminates and the summation 

 extends from r = to r = v, where 



n — m . 



v= — - — it n — m be even ; 



v= if n — m be odd. 



