PROFESSOR H. LAMB ON ELLIPSOIDAL CURRENT-SHEETS. 
157 
But 
(0 — ~ t; 2 + l’ 
g 9 '(fl = _ l K(C) 
g»co ^(C)? 2 (D(c 2 + i) z; 2 (?)’ 
and Jc 2 d(/dv l = rs/ly Hence, by (41), 
lh' (Co) 1 w;. 
£oPz (Co) J ^ 
47TCT 
IT 
& 2 Co (Co 2 + 1) arc cot 
t (C + iby« 3 ) 
=0 
|^y« 3 B 2 (/x) 
1 
2> 3 (Co)? 2 (Co)Co(Co 2 + 1) 
For the case of a c2is& we have, as in § 5, 
and 
<77 — 
ac 
x / (« 2 - r 3 )’ ^ = 
= C j 
where c is ultimately made = 0. Also 
Pi (°) = b Pi (°) = f > P/ (0) = 0, 
whence 
U 7 rcr 
K 7T y/ (a 3 — Ffi 
2 
U ( C + AJPya 3 ) - 
AY 
7 T y/ (a 2 — r 2 ) 
(2a 2 — 3r 2 ) 
7 r y (a 2 — r 3 ) 
U {C +py(2r 2 - a 3 )}.(94) 
The total charge on both surfaces of the disk is 
~ (C + K. 
The constant C is of course to be determined by the other conditions of the 
problem. If the axis of the disk be uninsulated, we shall have C = 0. 
16. The only terms in the value of ft which give rise to sensible currents in a 
rotating disk are those tessaral solid harmonics for which n — s is odd. 
O 
If the value of ft, referred to fixed axes, be 
ft = A . (1 — [jl 2 )* 12 
d'Fnfa) 
d/j, s 
(e +1)* 
d’p„ (C) 
dX* 
COS Sod, 
(95) 
