Prof.  J.  C.  Maxwell  on  Electric  Induction,  535 
This,  therefore,  is  the  value  of  the  magnetic  potential  of  the  current- 
sheet  at  any  given  point  on  the  positive  side  of  it.  Within  the  sheet 
there  is  no  magnetic  potential,  and  at  any  point  (£,  rj,  —  £)  on  the 
negative  side  of  the  sheet  the  potential  is  equal  and  of  opposite  sign 
to  that  at  the  point  ($,  77,  K)  on  the  positive  side. 
22.  At  the  positive  surface  the  magnetic  potential  is 
V=-J=2^ (3) 
At  the  negative  surface 
f=2** <4> 
The  normal  component  of  magnetic  force  at  the  positive  surface  is 
r~£~£ (5) 
dc,        dt,- 
In  the  case  of  the  magnetic  shell,  the  magnetic  force  is  disconti- 
nuous at  the  surface ;  hut  in  the  case  of  the  current-sheet  this  ex- 
pression gives  the  value  of  7  within  the  sheet  itself,  as  well  as  in  the 
space  outside. 
23.  Let  F,  G,  H  be  the  components  of  the  electromagnetic  momen- 
tum at  any  point  in  the  sheet,  due  to  external  electromagnetic  action 
as  well  as  to  that  of  the  currents  in  the  sheet,  then  the  electromotive 
force  in  the  directions  of  x  is 
_dF__  dxP 
dt       dx* 
where  \j,  is  the  electric  potential*  ;  and  by  Ohm's  law  this  is  equal 
to  <ru,  where  a  is  the  specific  resistance  of  the  sheet. 
Hence 
Similarly, 
<ZF_#  . 
dt       dx' 
r   .    .    .    .    . 
dG_dxP 
dt      dy'J 
(6) 
Let  the  external  system  be  such  that  its  magnetic  potential  is  repre- 
dP 
sented  by  —  -~-,  then  the  actual  magnetic  potential  will  be 
V«-|(P0+P), (7) 
and 
F=A(P,  +  P),      G  =  --^(P0  +  P),     H=0.    .     .     .     (8) 
dy  dx 
Hence  equations  (G)  become,  by  introducing  the  stream-function 
*  "  Dynamical  Theory  of  the  Electromagnetic  Field,"  Phil.  Trans.  1865,  p.  48.'}. 
