PROFESSOR H. LAMB ON ELLIPSOIDAL CURRENT-SHEETS. 
145 
II. 
9. The result of § 2 can be generalised, and it can be shown that the different 
normal types of free currents in a homoeoidal shell are obtained by equating the 
current-function cf> to the Lame’s functions of various orders. But it may be sufficient 
here to consider the case where two of the axes of the shell are equal, when the 
functions in question reduce to spherical harmonics. 
Taking first the case where the ellipsoid is of the prolate form, we transform to 
elliptic coordinates (£, /x, u) or (y, 0, a>) by writing 
x—k^/ (1 — /x 3 ) y/ (£ 3 — 1) cos o) = k sin 6 sinh y cos w 0 
y = k y (1 — /x 3 ) s / (£ 3 — 1) sin co = k sin 6 sinh y sin w >, . . . . (42) 
z — k (, [a — k cos 9 cosh y J 
the axis of z being that of symmetry. The value of /x may range from — 1 to + 1, 
that of £ from 1 to qo . The surfaces £ = const, are confocal ellipsoids of revolution, 
whose semi-axes are 
a — k — k s / (£ 3 —• 1) =-k sinh y, 
c = k £ — k cosh y, 
the distance between the common foci being 2k. The value of £ for the surface of the 
shell will be distinguished, where necessary, by £ 0 . The perpendicular on the tangent 
plane at any point of the shell is 
7 £ 0V /(£ 0 *-l) 
— fC ; o ox 
Laplace’s equation v 3 V = 0 transforms into 
d \ . _ dV] , 1 d? Y 
d/xffi 1 ^ dp\ + 1 -y dco* 
1 d 2 Y 
+ 1 - £ 2 rf © 3 ' 
(44) 
Considered as a function of /x, o>, Y may be expanded in a series of spherical 
harmonics whose coefficients are functions of £, and it is easily seen that each term of 
the expansion must separately satisfy (44). Taking first the case of the zonal 
harmonic, if we put 
v = p M (/x).z, 
where 
p « (/*) = 
1.3.5. ,(2xi-l) 
i n 
/x" — 
we find 
MDCCCLXXXVII.-A. 
n(n-l) 
2(27i- 1 )^ 
n(n - 1) (n - 2) (n - 3) _ 
~ t ~ 2.4 (2n — 1) (2 n - 3) ^ 
Lf{ (1 -^) |} + U» + i>Z = °, 
...}, (45) 
• • (46) 
u 
