158 Dr. T. J. Fa. Bromwich on 



The function ^„(«), used by Lamb*, is connected with S n (z) 

 by the equation 



S re (z) = z"+^ n (z\ (5-52) 



as may be seen by comparing the series for \p n (z) with (5*5). 

 Similarly we have 



= Cn(z)~&(z), (5-6) 



where 



°.<->-*'(-i!H™*) 



1.3..,(2n-l) f * . 



s» \ 2(1— 2ny2. 4(1 -2n)(3 -2n) 



-...}. . (5-61) 



In terms of the modified Bessel function K n , we find thatt 



E^) = v /^)^ + *^K M+i (^). . . . (5-62) 



For divergent waves Lamb uses a function f n (z) which is 

 connected with E„(z) by the equation 



E n (z) = z»+\f n (z) (5-63) 



He writes further 



f n {z) =Vn{z)-^ n (z), 

 so that 



C n {z) = z"+i* n (z) (5-64) 



Difference relations amongst the functions S», C w , E n . 



These relations are the saine for all three functions, but will be 

 stated for the S»-f unction only. 

 From (5-5) we have 



S B+l (s) = - z » + i|{M^} =!i±is„(--)-S/( z ). (5-7) 

 Again, it will be seen that 



/* +-V+P = (2n+p)z"+P-\ 

 and if this result is applied to the series in (5'5) we find that 



(£+=)*« = *^*) 



or S>-i(*)= jS^) + S»'(*). • • • (5-71) 



* ' Hydrodynamics,' 1906, Art. 287. 



t And thus ~E n (z)=v — m in the notation of ]\Iacdonald's paper quoted 

 above, so that n=C n (z). 



