154 
PROFESSOR H. LAMB ON ELLIPSOIDAL CURRENT-SHEETS. 
where 
V (0 = (C* + 1) 
<(0 = (£ 8 + i) 
sii d *MQ 
dp ’ 
j /2 
<z(? 
When w = 1, (81) becomes 
Co 
t Co (Co 2 + 1 ) 2 {arc cot Co~ ^ a+1 j > 
or, putting 
k\/ (Co' +0 — a > Co' + 1 —l/ e b 
27rah f v /(l - e 3 ) . 1-e 2 
r = -<1 --s-arc sm e — 
. (83) 
which agrees with (13). 
13. If we make £ 0 infinitesimal, we get results applicable to a circular disk, of the 
kind considered in § 5, provided n — s be odd. Putting k = a, kt, 0 = c, and 
p/e = 2 //ot — 2 Pq'c, 
where c is ultimately made to vanish, the formulae for the symmetrical currents 
become 
= C.P„ 1 , 
7-2 \i ' 
(84) 
inra ■ 
n(n+l)p 0 '[ Pu ^ q ’ 1 
■ ( 85 ) 
where r denotes the distance of any point of the disk from the centre, 
values of £ we have, n being odd, 
For small 
p» (0 = 
1 . 3 ... TO 
/ ir , 
— o P« 
whence 
2.4 . . (to - 1) 
% + 1 nr 2 a f 1 . 3 ... to 19 
r = 
For n — 1 , this gives 
to p,/ [2.4 .. . (to + 1) 
r = n 2 a/2p 0 ', 
(86) 
as in § 5. The persistencies of the successive symmetrical types are as 
1 — —— 
1 9 8 > 6 45 
