PROFESSOR H. LAMB ON ELLIPSOIDAL CURRENT-SHEETS. 
147 
is a zonal, tessaral, or sectorial harmonic of integral order. Since any arbitrary 
value of (f> can be expanded in a series of such harmonics, the results thus obtained 
will enable us to represent the decay of any initial distribution of current whatever. 
If ds d.y, be linear elements drawn on the surface along a meridian and a parallel 
of latitude respectively, viz., 
- i*) 
dfx 
ds, = JV(to* - l ) \/(! 
(50) 
the current may be resolved into the components 
d<f> 
along the meridian, towards the positive pole ; and 
along the parallel, in the direction of oj increasing. 
CCS,, 
cK 
d(f> 
Take, first, the case of the zonal harmonic 
<f> = C . P„ (p.) 
(51) 
The currents then flow in circles round the axis of 2 , the strength of the current at 
any point being 
C 
G7 
&W(Co 2 - 1) 
v/(l ~fS) 
d P„ (fi) 
d/x 
(52) 
If n be the magnetic potential due to these currents, we have 
V 2 n = 0, 
with the conditions that at the surface £ 0 
n 1 — xi 0 -f- 47 T(f), 
(the suffixes denoting the values on the two sides), whilst the normal derivative is 
continuous. Assuming 
n 0 = AP„(p)P„(£) "I 
n 1 = BP4 / x)Q ;i (0j ’ 
the surface conditions give 
B Q* (Co) — A P« (Co) + 4 ttC, 
bq;(Co) = ap ? /u 0 ); 
whence, in virtue of the relation 
p: (0 q. (o - p. (o q; (0 = , 
U 2 
(54) 
