152 
PROFESSOR H. LAMB ON ELLIPSOIDAL CURRENT-SHEETS. 
the electric potential xjj. It may be worth while, however, to record the value of \p 
thus obtained for the space included between the two surfaces of the film, viz. :— 
w 
here 
Xp = D (3£ +1 — ^_ 1 ) cos sco + E (UUi — Ui_j) cos SCO, 
— 1 11 ~ S ( i _ 2y/2 / 5*2 _ ] y/2 (0 
^ d ^ (4 a? 5 
Ip _ \ U - S 1 1 _ 2y/2 /v 2 _ y/2 d^QJX) 
jrc + s^ ^ ) dfi* 14 ’ dX s ’ 
(73) 
and D, E, are certain constants. The first part of (73) corresponds to a distribution 
of electricity on the outer surface of the film, the second to a distribution on the inner 
surface. In the case of the sectorial harmonic, E = 0. 
12 . When the shell is of the oblate form, the elliptic coordinates to be employed 
are as follows. We write 
x — lc */(1 — /E) \/(£ 2 + 1) cos co 
y—k y/{\ — p 3 ) v/(£ 2 + 1) sin w 
= k sin 6 cosh rj cos co 
= k sin 6 cosh rj sin co 
>, ■ 
z = &p£ 
= k cos 9 sinh r\ 
(74) 
where £ nray range from 0 to co, The surfaces £ = const, are confocal ellipsoids of 
revolution, the extreme case £ = 0 being a circular disk of radius k in the plane xy. 
Comparing these equations with (42), we see that they may be obtained by writing 
i £ for £ and — ik for k. The equation V 2 V = 0 therefore becomes 
±\ t ., 9A dV] , 1 d?Y 
rfpC 1 ^ ) dfi\ + 1 - fj? dco* 
d 
dV 
d£ ] ^ "I" ^ dt I + 1 
dK 
i d~ v 
+ ?dco~' 
• (75) 
The type of solutions symmetrical about the axis is 
V — P„ (p). Z, 
where 
d 
di {(l + ^) li \-n(n+l)Z = 0. 
■ ■ ■ (76) 
One integral of this is 
Pn(C) = 
1.3.5... (2n — 1) 
Iff 
l>„, ^0-1) n(n-l)(n-2)(n-3) 1. 
l 4 + 2(2?i-l) 4 2.4 (2n — 1) (2n — 3) 4 +•-•}’ (") 
* Pee Ferrers, ‘Spherical Harmonics,’ chap. vi. 
