CURRENTS IN INFINITE PLATES AND SPHERICAL SHELLS. 
317 
the equations (13) transform into 
dH 
d.pG 
dfl 
pdcf) 
pdz 
~T P 
dF 
dH 
dn 
dz 
~ dp 
pdcf) 
d.pG 
dF 
dn 
pdp 
1 
CL 
1 
dz 
(18) 
Here F, G, H are now the resolved components of F, G, H in the p —, (f) —, z — 
directions, and the equations are obtained by resolving the two sides of equations (13) 
in these directions. The expressions on the left hand sides of the above equations 
are most readily recognised by observing that, when they vanish, we must have 
[ F c lx + Gcly +Hdk=] Fdp + Gpdcf) + H dz 
an exact differential. The equations themselves are satisfied by 
V 3 P=0, 
ldP dP 
'pdfi J ~ + dp’ 
> 
H = 0 
and the condition of no convergence 
o 
(19) 
1 d.pF , dG dH . . , 
-=0, is satisfied. 
p dp pdcf) dz 
(■c.) If we employ jiolar coordinates r, 6, (f> 
\Fdx-\-Qdy-\-Hdz]=~Fdr-\-Qrd9-\-H.r sin 0d(f) 
and the results of transforming the coordinates are obviously 
__1_ ( d.r sin dH _d.rG\ 
r 2 sin 6 \ d9 d<j) ) 
1 /dF d.r sin 0H \ 
r sin 6 \d<fr dr ) 
1 I d.rG dF\ 
r \ dr d6 ) 
These equations are satisfied by 
2 T 2 
dll 
dr 
m 
rd9 
dVL 
r sin 0dcp 
. . . (20) 
