182 DK. T. J. I'A. BKOMWICH ON THE 



first expression in (3'l) ; while the latter is found by combining the two expressions 

 so as to yield 



(3'2) ' \j.frJ r 



Using the notation explained in (3'4) and (3*5) below, the standard functions are 



E B (/c,r) for divergent waves, 



and 



S n (iqr) for waves within a spherical boundary. 



Consequently, for waves inside a spherical boundary, (2'5) and (2'8) can now be 

 replaced by the forms 



f U or V = 2S.(/c I r)Y,.(fl, <f>), 



( 3 ' 3 ) v n(n+l) g / x v /, A 

 A or ca = 2, 3 ' Ql*i*v -J-n W> #/> 



L_ 



for divergent waves the function S n (K t r) must be replaced by E n (*!?). 



Definitions and Properties of the Two Standard Functions S n (z), E B (z). 

 We write for brevity 



1- 



" 1 . 3. 5...(2n + l) I 2(2w+3) 2 . 4 (2 

 In terms of the known Bessel function we can write 



(3'41) S,(z) = 



and accordingly the function S n (z) is the same as that denoted by u in one of 

 MACDONALD'S papers.* 



In the notation adopted by LAMB,! and those writers who have used LAMB'S 

 solutions as the fundamental forms, we have the identity 



(3-42) S.( 



* 'Phil. Trans. Eoy. Soc.,' A, vol. 210, 1910, p. 113. See in particular p. 115. 



t ' Hydrodynamics,' 1906, Art. 287. 



