534  Royal  Society  £*** 
city  which,  in  unit  of  time,  crosses  from  right  to  left  a  curve  drawn 
from  a  point  at  infinity  to  the  point  P. 
This  quantity  will  be  the  same  for  any  two  curves  drawn  from  this 
point  to  P,  provided  no  electricity  enters  or  leaves  the  sheet  at  any 
point  between  these  curves.  Hence  <£  is  a  single- valued  function  of 
the  position  of  the  point  P. 
The  quantity  which  crosses  the  element  ds  of  any  curve  from 
right  to  left  is  ^ 
-^as. 
as 
By  drawing  ds  first  perpendicular  to  the  axis  of  x3  and  then  per- 
pendicular to  the  axis  of  y,  we  obtain  for  the  components  of  the  elec- 
tric current  in  the  directions  of  x  and  of  y  respectively 
u=%  t>=-^ (1) 
ay  ax 
The  curves  for  which  <p  is  constant  are  called  current  lines. 
20.  The  annular  portion  of  the  sheet  included  between  the  current 
lines  0  and  <p-\-S(j)  is  a  conducting  circuit  round  which  an  electric 
current  of  strength  t)</>  is  flowing  in  the  positive  direction — that  is, 
from  x  towards  y.  Such  a  circuit  is  equivalent  in  its  magnetic  effects 
to  a  magnetic  shell  of  strength  dtp,  having  the  circuit  for  its  edge*. 
The  whole  system  of  electric  currents  in  the  sheet  will  therefore 
be  equivalent  to  a  complex  magnetic  shell,  consisting  of  all  the  simple 
shells,  defined  as  above,  into  which  it  can  be  divided.  The  strength 
of  the  equivalent  complex  shell  at  any  point  will  be  (j>. 
We  may  suppose  this  shell  to  consist  of  two  parallel  plane  sheets  of 
imaginary  magnetic  matter  at  a  very  small  distance  c,  the  surface- 
density  being  -  on  the  positive  sheet,  and  —  -on  the  negative  sheet. 
21.  To  find  the  magnetic  potential  due  to  this  complex  plane 
shell  at  any  point  not  in  its  substance,  let  us  begin  by  finding  P,  the 
potential  at  the  point  (£,  y,  £)  due  to  a  plane  sheet  of  imaginary 
magnetic  matter  whose  surface-density  is  <j>,  and  which  coincides 
with  the  plane  of  xy.     The  potential  due  to  the  positive  sheet  whose 
surface-density  is  Z,  and  which  is  at  a  distance  \c  on  the  positive 
side  of  the  plane  of  xy,  is 
That  due  to  the  negative  sheet,  at  a  distance  \c  on  the  negative  side 
of  the  plane  of  xy,  is 
¥*+*•*+*> 
Hence  the  magnetic  potential  of  the  shell  is 
V=-~ (2) 
d£  K  ' 
*  W.  Thomson,  "  Mathematical  Theory  of  Magnetism,"  Phil.  Trans.  1850. 
