CURRENTS IN INFINITE PLATES AND SPHERICAL SHELLS. 
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clp and dcf> respectively, the directed quantities, as above, preserving their former 
designations. 
Our equations are 
1/dy d p/3\ da dy 1 (d p/3 da. 
47T/ '~pV^> dz)’ 7TV= lz~Jp 7T ' C= d\Jp~d^ 
i/dn_d P G\ of 
11 /I JL rU ’ P 
r/F r/H 
p\d<f> dz )’ 
and we may satisfy these by 
dz dp’ 
V z 
1 (dpG dF 
p\ dp dcf) 
and 
F = — 1 G='y, H = 0 
p d<p dp 
dr P _ 1 d 3 P Id / dF\ , 1 d 3 P- 
» P=—ZJFTr 7 =~tApT-I+~2 
dpd. 
p def>dz’ ' p dp\ dp) p“ dfi 2 
1 d<E> 
u= - v=~, w= 0 
p d(p dp 
(23 x ) 
• • • (23*) 
here 
4tt(P= — V ~P ; 
d 1 d 2 d 2 
p dp Pdp p 3 dcf > 3 dz 3 
(c.) Polar coordinates. 
The equations in this case are, resolving along dr, rdd, r sin ddef), 
1 [dy sin 6 d/3\ . 1 (da dry sin d \ 1 fd.rB da 
4 ttu = ——[—- -, 47ry=—^ tt—-;- , 4 tt^= 
r sm d\ dd ref) ) r sin 0\d<f> dr ) 
1 /dH sin 6 dG\ 
r\ dr d6 
CL — 
r sin 6\ dd d<f>)’ 
and we may put in these 
F=0, G=- 
1_ d P TT_^ > 
sin 6 deft’ dd 
(24i) 
a — -f- 
r 
‘ 1 d . „dP 
yin d dd ' 8111 dd 1 sin 3 0 d<p 2 _ 
U-=z 0 , V — 
1 d 3 P 
d<f> 
’ P 7- drdd ’ 7 
d<3> 
sin dd6’ W dd 
where 
1 d 3 (Pr) 
r sin d drdcf) 
y • ( 24 *) 
j 
4tt<P= — V ~P. 
