166 
MR. J. H. JEAXS OX THE VIBRATIOXS AXD 
Hence expression (33) becomes 
2 [CWJ„_.(Kr) - + (2u + l)(n - 1) Dr’’-’J„_jKa)], 
of which the value at r = a is 
^^i = 2aiC/c-J,,_,(/ca) + 2[{2u + l)(n - l)Da'^-^ - na-K^K\.].,_,{Ka). 
Phis IS the value at t = a of the term which occurs in curled brackets in 
equation (31). The value of the similar term in (32), namely expression (34), is seen 
to he 
do — 2«-C/c‘J„+^ (/c«) — ‘2 (n-j- 1) (/ca) 
Write 
2:1 — [ko) . . . 
r = A + ^ («,) . , , 
fJb -r- \ / 
then eijuatmus (31) and (32) become, at r = rq 
a + «, + a—A. (,-v. + (2« + 1) (<■ 
(Ir 
,1 
<m 
d + (^' ^Vq,) — (2u q- 1) ( r j = U 
Now Ave have, from equation (26), 
(35). 
(361, 
(67), 
P)=0 . (38), 
• . . . (39). 
Write 
dr L, 
I I r ^ 
-i- AR-+ (aoh =« 
{'In + 1) p- 
drrpn _ /d(2/i 3-'1 ) 
{In + 1 ) p~ _ {'hi + l)yr — 47 rpii. 
then this last ci^uatioii becomes 
Now, at r — «, 
a„ = C' rxc 
i/Q'i 
or 
II, 
'I 
ih 
.('■'"A) = ('‘+ 2 )a„_« 
fh 
a 
dr 
and equations (38) and (39) become 
;t,y I - 2)r / ^ 
a + —, (a„ + ^ 1 
r \ ■ dr 
da, 
V — {yi 1) + a — (2n + 1 
m which r must be put equal to a. 
dr 
dp 
dr 
- P ) = 0 . (40), 
• • (61), 
