SMALL VIBRATION'S OP A HOLLOW VORTEX. 
181 
In finding the second part it will be well for the later approximations to carry i Jj a 
term further to include A a . Then from (7), if U 3 be the part of U due to this 
where 
^v/(C-e){Bo+SB„ cos nv } 
P P 
H —_lA b__ 3 A Al 
— g-^Op/ Wlp/ 
xt o Xt i 
p> — 1 A 1 >fl 3 A (Po + P 2 ) , 
P 
15A A- 2 
4 
L n Xfc ii+l J L Xl/ XV. n J 
These values of U l5 U 3 are to be expanded in a series of cosines of multiple angles. 
But here it is only needful to keep terms of the same order as A a , or compared with 
Now 
A 0 of order k~ 
2C=Jc+ : . 
Hence 
1—C c _/I \/ 1 cos® cos2» 
C-e — \C _ C Jr‘*‘2C i “' cT + “20 
cos 2v 
2C 4C 3 
cos 3v 
= &(1 — & 2 ) — (1 — k 2 ) cos v—h(l —W) cos 2v —P cos 3v 
Also 
v/2(C—c)*{B 0 +2B» cos nv} 
= ^ 2C { 1 } ( B 0 + B 1 C0S 7; + B C C0S 2l ’+ B 3 C0S 3V ) 
=^p{l-P 2 -^l-F)cos'y-iPcos2v}{B 0 + . . . } 
= {l +^& 3 —h cos v—ffi cos 2-y} {Bq+Bj cos v+Bo cos 2v}Jc~ i 
= {1 + i^ 3 —& cos v—ffi cos 2v } (B 0 + B x cos v)Je~*-\-( 1 — & cos v)B 2 k~ h cos 2v 
considering at present B 0 and B x to be of the same order. From this it is easy to 
show that if 
