SMALL VIBRATIONS OF A HOLLOW VORTEX. 
171 
But 
2C^-'-SH,= K-i){2CSP„-|(P, +1 -P„_ 1 )} = K-i)S(P, +1 + P,- 1 ) 
Hence 
^~ 2(C- ~+{ -W-I^Pi + iAoPo COS v+Zfi* cos nv j ... (7) 
where 
B n ={(v?— i)A„-(w+l] 3 —i)A„ +1 }P w+1 -{(?i—l| s — (n z — 
4. For reference I here insert the values of the first five orders of the functions, 
expressed (a) exact in terms of the elliptic integrals, and (/3) approximate in a series 
of ascending powers of the modulus. Throughout this paper the moduli k, 1c are used 
instead of the kf, k of the paper on Toroidal Functions. It has been thought advisable 
to do this as all the approximations go according to powers of k (the old k'). Hence, 
of course, E, F appear in place of E', F', and vice versd. 
We know that P^=a«E / +/3„F / , Q„=a„(F — E) + /3„F. 
Hence for the first set of formulae we require only to tabulate a„, For the first 
five they are 
a 0 =0 
cq= 2k ~“ 
a 3 =-f(l+P)& - * 
a 3 =^ 5 -(8 + 7P+8^)^ 
“*=3X7 ( 6 + 5 ^+5**+ 6 ^- S . 
A=2i* 
A=° 
3.5.7 
(24P+23& 4 +24F)7r* 
^ ( 8 ) 
a . = ^(128 + 104P+99^+104T; 8 +128F)7rq #.= 
■— (16^+15^+15^ 
+ 16P+ 
These are exact. In the applications which follow 7c will nearly always be a very 
small quantity, so that a few terms of the series will give the values very approxi¬ 
mately. By substituting their values for E, F, E', F' in terms of k, the expressions 
4 
become, writing L for log - 
rb 
P 0 =2{L+i(L~l)F+/ 4 -(L-f)^+^ 6 -(L-|^+ 
P 1 =2{l+i(L-i)F+*(L-if)^+i 1 ^(L-t)F+ . . . }Jr* 
P 3 =«l+P a +A(L-A)**+M(L-^)* B + • • • }k~i 
P 3 =if{l+P 3 +M^1 1 ^(L-M)F+... }k-i \ 
P 4 =ff{i+^ 3 +M^+^ 6 + • • • }*-* 
953 
p 5= ^ U +^+^+^ 6 + ... }*-* 
