Expressing Solutions of Laplace's Equation. 209 



For consider 



H^^'-f^£)^'-^y' 



If.. 



l-f^V 2 ^ 



pp;;'+-"-"(_l)>-+^,! 





V PS(-l)' 



5 — 0- 



5 ! (W 5) ! 



-svhere p = In — . \ i£ ni be even, 



p =yn — j i£ m be odd, 



a = -^ i£ ?>? be even, 



cr = — — i£ 7)1 be odd. 



Hence 



18. It might be thought that analogous symbolic ex- 

 pressions could be obtained for the Q-functions. 

 Thus we know that i£ 



J ^ I- 2 fit -^f 

 taken round a contour surrounding the two branch -points 

 t=fi±\/ ir—l. then 



q,^\ ^-—JL==^.dt 



taken round a figure of eight, each of whose loops incloses 

 one o£ these branch-points. 



Now in the case of the first contour 



PliiL Mag. S. 6. Vol. 6. No. 32. Aug. 1903, P 



