160 Dr. T. J. Fa. Bromwich on 



§ 6. Comparison of the formulce of § 5 with those given 

 by Lamb. 



Prof. H. Lamb's solutions * are given in Cartesian form, 

 and have formed the starting-point for a large number of 

 investigations on electromagnetic waves, in spaces bounded 

 by spheres. 



In view of the general proof of § 2 that the solutions (3"1), 

 (3*2) include all solutions suitable for spherical problems, it 

 is evident that Lamb's solutions must be included as well as 

 others. But a direct proof is not without interest, as the 

 complete verification is less obvious than might perhaps be 

 expected ; the work depends on the difference-relations, 

 (5'7)-(5*72), obtained above for S»(V). Lamb's solutions 

 are divided into two classes, corresponding to the two 

 functions U, V employed in equations ' v 3"l) and (3'2). All 

 these solutions contain the simple harmonic time- factor e 0KC \ 

 so that they can be compared directly with those given in 

 (5-3) and (5*4). 



For simplicity of statement, let us take K=l, /i = l, so 

 that *c 1 = *c. 



Lamb's solution of the " first class " has a Cartesian component 

 (parallel to #), 



where 



W = r\b n {Kr)xa- 



Here \ n is a solid harmonic of order n, aud so is capable of 

 being regarded as a multiple of r n Y n (d. <f>) : thus we may write 



W = AS n ( K r)Y n (0, <p). 



Now the vector of which (6*1) is the ^-component is the 

 vector product of the two vectors 



Translating to spherical polar coordinates, the two vectors (6-11) 

 become 



(1,0,0) and (- — ____,. — ) . . (6-12) 



v ' ' J \ dr r 30 r sin d0 / v ; 



* Originally given in Proc. Lond. Math. Soc. vol. xiii. p. 51 (1881) ; 

 here we shall quote results given in Art. 335 of Lamb's ' Hydro- 

 dynamics' (1906), where other applications of the formulce are referred 

 to. It should perhaps be noted that the function denoted by 4' n { z ) has 

 a different numerical factor in Lamb's earlier work ; thus the formulae 

 corresponding to (6-3) are somewhat different in appearance in the 

 earlier papers. 



