PROFESSOR H. LAMB ON ELLIPSOIDAL CURRENT-SHEETS. 
155 
For the tessaral harmonic 
where 
we have 
^ r s d s P„ (yu.) . * 
(h = (J — —--sm Sco*. 
p a* d/i* 
(87) 
tli 
1 
dt n s 
du,f 
Po \n +8 
n(n + 1) — s 2 
_1 
dX] 
<r=o 
From the series (77) we find, without difficulty, 
1.3... (to + s) 
dtn_ 
AS 
f = o 2.4 ... (to — s — 1) 
n — s being odd. Also from the second line of (78), under the same restriction, 
du n 5 _. \ 7T dt H s 
AY “ ' A ’ 
when £ = 0. Hence 
7T"« 
ITO — S 
(to (to + 1) — s 3 }pf + s [2.4 ... (to — s — 1) 
.1 1 • 3 ... (to + s) 1 3 
{ 2 .4... (to — s — 1)J.' 
The most important of the types (87) is that in which n = 2, s — 1, when 
Q 9 / / 
t = 10 ^ . 
14. The methods of §§ 10, 11, might be applied to determine the currents induced, 
by simple harmonic variations of a magnetic field ; but it is unnecessary to go through 
the calculations, as the result can be written down at once from the following 
considerations. 
We must first suppose the magnetic potential (12, say) due to the field to be 
expanded, for the space near the conductor, in a series of terms of the forms given by 
the first lines of (47) and (49)t; or of (79) and (80), as the case may be. Each of 
these terms will act by itself, and produce a current-function </> of the type (51) or 
(64). Now, in § 11, the equation of free currents of any normal type was brought to 
the form 
W> = J f t , .(89) 
* The current lines are the orthogonal projections on the plane xy of the contour lines of spherical 
harmonics drawn on a sphere of radius a. 
f F. E. Neumann has shown (‘ Crelle,’ vol. 37) how to expand the potential of a single magnetic pole 
in this way. 
