532  ,  Royal  Society : — 
gative  direction,  so  as  to  include  all  the  images  which  have  been 
formed  in  all  past  time,  the  magnetic  effect  of  this  imaginary  train 
at  any  point  on  the  positive  side  of  the  conducting  sheet  will  be 
identical  with  that  of  the  electric  currents  which  actually  exist  in 
the  sheet. 
Before  proceeding  to  prove  this  statement,  let  us  take  notice  of  the 
form  which  it  assumes  in  certain  cases. 
10.  Let  us  suppose  the  real  system  to  be  an  electromagnet,  and 
that  its  intensity,  originally  zero,  suddenly  becomes  I,  and  then  re- 
mains constant.  At  this  instant  a  positive  image  is  formed,  which 
begins  to  travel  along  the  normal  with  velocity  R.  After  an  interval 
St  another  positive  image  is  formed  ;  but  at  the  same  instant  a 
second  negative  image  is  formed  at  the  same  place,  which  exactly 
neutralizes  its  effect.  Hence  the  result  is,  that  a  single  positive 
image  travels  by  itself  along  the  normal  with  velocity  R.  The  mag- 
netic effect  of  this  image  on  the  positive  side  of  the  sheet  is  equivalent 
to  that  of  the  currents  of  induction  actually  existing  in  the  sheet ;  and 
the  diminution  of  this  effect,  as  the  image  moves  away  from  the  sheet, 
accurately  represents  the  effect  of  the  currents  of  induction,  which 
gradually  decay  on  account  of  the  resistance  of  the  sheet.  After 
a  sufficient  time,  the  image  is  so  distant  that  its  effects  are  no  longer 
sensible  on  the  positive  side  of  the  sheet.  If  the  current  of  the 
electromagnet  be  now  broken,  there  will  be  no  more  images  ;  but  the 
last  negative  image  of  the  train  will  be  left  unneutralized,  and  will 
move  away  from  the  sheet  with  velocity  R.  The  currents  in  the 
sheet  will  therefore  be  of  the  same  magnitude  as  those  which  fol- 
lowed the  excitement  of  the  electromagnet,  but  in  the  opposite  di- 
rection. 
1 1 .  It  appears  from  this  that,  when  the  electromagnet  is  increasing 
in  intensity,  it  will  be  acted  on  by  a  repulsive  force  from  the  sheet ; 
and  when  its  intensity  is  diminishing,  it  will  be  attracted  towards  the 
sheet. 
It  also  appears  that  if  any  system  of  currents  be  produced  in  the 
sheet  and  then  left  to  itself,  the  effect  of  the  decay  of  the  currents,  as 
observed  at  a  point  on  the  positive  side  of  the  sheet,  will  be  the  same 
as  if  the  sheet,  with  its  currents  remaining  constant,  had  been  car- 
ried away  in  the  negative  direction  with  velocity  R. 
12.  If  a  magnetic  pole  of  strength  m  be  brought  from  an  infinite 
distance  along  a  normal  to  the  sheet  with  a  uniform  velocity  v  towards 
the  sheet,  it  will  be  repelled  with  a  force 
4z2K+v 
where  s  is  the  distance  from  the  sheet  at  the  given  instant. 
This  formula  will  not  apply  to  the  case  of  the  pole  moving  away 
from  the  sheet,  because  in  that  case  we  must  take  account  of  the 
currents  which  are  excited  when  the  pole  begins  to  move,  which  it 
does  when  near  the  sheet. 
13.  If  the  magnetic  pole  move  in  a  straight  line  parallel  to  the 
sheet,  with  uniform  velocity  v3  it  will  be  acted  on  by  a  force  in  the 
