PROFESSOR H. LAMB ON ELLIPSOIDAL CURRENT-SHEETS. 
149 
Since 
this gives, for n = 1, 
E d) = t, 
2 ~° S f - 1 
Q. (£) = I log ID - i. 
T = ~ & & ~ 1)M ^ - i l0g 
2 J _£> ] 1_So + 1 
P 
fo 2 -l 
5,-iJ ’ 
or, putting 
a = k v /(C 0 2 - 1), C 0 = 1 / e > 
2nrede f 1 1 — e 2 , 1 + e 
T =— <--^r l °sr^ 
which agrees wdth (13). 
For n= 2, I find 
T = 
2irct 2 e f 3 - 2e 2 „ 1 - e 2 , 1 + e 
— -:— iOff- 
e 
.4 2 ^5 S> i — g 
(59) 
• (60) 
The case of a spherical shell may be deduced from (58) by making k = 0, £ = co , 
&£ = a. For large values of £ we need only retain the first terms of the series in 
(45), (48). In this way we reproduce the known result 
47r« 2 e 
T = (2 n + 1) p 
11. The simplest plan of dealing with the case where the current-function is a 
tessaral or sectorial harmonic is to consider the current round any infinitely small 
circuit bounded by meridians and parallels. If ft, S, be the components of electric 
momentum along a meridian and a parallel respectively, we have 
or, putting 
/ dcf) 
dR 
di|r 
P ds a ~ 
dt 
d.s> 
n , dcf, _ 
dS 
dyjr 
P ds „ 
dt 
~ds„ 
P f 
> A# 
-a 
>; 
eel he £ 0 \/(T 0 2 — 1) 
£ 0 2 — pP dcf, dR ds, 
P_ . _ _ 
he £ 0 (£ 0 3 -l)(l -pP)dco' 
_ (h/O 
dt dp dp I 
p , 2 n dcf, _ dS dSu d\jr 
jfce? 0 ’ ^ djl ~~ ~ ~di dco ~ dco 
Eliminating xp, 
d 
he£ 0 L dp [ 
j(i-r)tb + 
dp 
r 0 3 - ^ 
d 2 f 
(£ 0 2 -m-/) dco\ 
dt [do \ dp 
d /g4 
dp, \ d co 
(61) 
(62) 
( 63 ) 
