SMALL VIBRATIONS OP A HOLLOW VORTEX. 
169 
or 
— 2A 0 -f 
2 ctct m 
"VS 
p' v 
-L m *4 
r=m +1 
P' P' 
i r-L j—i 
(n > m) 
A„=2A 0 (n<m) 
The co-efficients are now determined to the extent of one arbitrary constant. This 
appears because 4> is also indeterminate to the extent of an additive constant. As 
this constant is expansible in a series of the form \/{Q — c)%K n cos nv, it introduces 
the undetermined constant A 0 above, which must be determined by the condition that 
the series must be convergent. This cannot be unless A oo =0, which requires 
A n 
aa m V' °° 
■v/s m+1 p;.p / ) .. 1 
whence 
* _ 2<xoc m P m 1 
A„= 
x/S ^ +1 PVP / r _ 1 
2<X0C m P'„, 00 1 
- ni ^ 
v/S ^ +1 P , ,Pb_ 1 
(n>m) 
(n<m) 
. . . (5) 
So also the terms in sin nv will give 
2 a 
(C -c) 
*/W=“S 1 {(B,~ B„ +1 )F 
(B*_!—sin nv 
C-cA 
and the particular case f(v) = [ —) (3 m sin mv produces the same equations as before, 
except that the last is 
whence 
x 2 —£^=0 where £C 1 =B 1 P / 0 P , 1 
B.=B l PVP'A -n_+^p',£ +1 —L- (»>m) 
r r r V ° r r x r -1 
—B X P 0 P 1 S 1 p, p , (n < m) 
x r ± r-i 
and the condition of convergency determines B 1; so that 
MDCCCLXXXIV. Z 
