182 
DR. T. J. I’A. BROMWICH ON THE 
first expression in (3'l); while the latter is found by combining the two expressions 
so as to yield 
. 1 / 1 3Y sin(/cir) 
r drj r 
(3-2) 
^n +1 j_ 
Using the notation explained in (3‘4) and (3‘5) below, the standard functions are 
and 
(«-p’) for divergent waves, 
S„ (/fir) 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 
(3-3) 
U 
X 
or 
or 
V = 2S. M Y. (9, i,), 
for divergent waves the function S„ (k^v) must be replaced by E„ (kit). 
DeJi 7 iitions mid Pro^ierties of the Two Standm'd Functions S„(2), E„(2). 
We write for brevity 
I.-.) 
” 1.3.5... (2n+l) r” 2(2n + 3) ^ 2 . 4(2w + 3) (2n+5) 
In terms of the known Bessel function we can write 
(3-41) S,(2) = .\/(|)j.,,(2), 
and accordingly the function S„ (2) is the same as tliat 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 
S, (z) = Z” 
* ‘Phil. Trans. Roy. Soc.,’ A, vol. 210, 1910, p. 113. See in particular p. 115. 
t ‘ Hydrodynamics,’ 1906, Art. 287. 
(3-42) 
