PROFESSOR H. LAMB ON ELLIPSOIDAL CURRENT-SHEETS. 
151 
the reduction being effected by the formula 
tj .? _ rr _ /_y l!L±J 1 
” dX * dX 1 ’ j n-s X 2 - 1 
(70)=* 
# 
Hence the right-hand side of (63) 
d dn 
dt ~dX’ 
7- IT i \ (a 2 \«M!d J P« (A 6 ) • 
- ~ 4 - x > m 0 ~ 'U -^r sm *“• 
= (-r 1 4 ^ (C -1 r ■ 1,4 
dto dt ;0 c77 
(71) 
by (64) and (69). Equating (65) and (71), we see that the assumption (64) satisfies 
the conditions of a normal type, and that the corresponding modulus of decay is 
4ttF6 I n-_s roOTo 2 - l) 3 dTX tf 0 ) dU n '(X 0 ) 
p |n + s n(n + 1) (f 0 3 — 1) + s 2 d£ 0 d£ 0 
(72) 
The accuracy of this result may be tested by putting k — 0, £ 0 = co , /c£ 0 = a, when 
we obtain the correct result 
477Vf 2 e 
(2 n + 1) p 
for the case of a spherical shell. 
The case of the sectorial harmonic is obtained by putting s = n in (72). When 
n = 1, in which case <£ o c y, we obtain 
or, writing 
2rf e ? 0 2 (r 0 3 -1 ) 2 rr 0 , ,.?o + i 
T =— 2ir=-i ' l2 lo «?“i 
y- 2 1. 
?0 2 - 1 J ’ 
Co = 1/e, & 2 (C 0 2 — 1) = « 2 , 
27ra 2 e 1 
p 2 — e 2 
1 - e 2 . l+e 
2e 3 ° g 1 - e 
which will be found to agree with (14). 
The results (58) and (72) were originally obtained by a method more analogous to 
that of § 2, the currents and the electric momentum being resolved parallel to x, y, z. 
This method is much longer than that here given, and involves the determination of 
* Todhuntek, ‘ Functions of Laplace, &c.,’ § 109. 
