530  Royal  Society : — 
of  any  magnet  or  electromagnet,  a  change  takes  place  in  the  mag- 
netic field,  electromotive  forces  are  called  into  play ;  and  if  the 
material  in  which  they  act  is  a  conductor,  electric  currents  are  pro- 
duced. This  is  the  phenomenon  of  the  induction  of  electric  currents, 
discovered  by  Faraday. 
I  propose  to  investigate  the  case  in  which  the  conducting  substance 
is  in  the  form  of  a  thin  stratum  or  sheet,  bounded  by  parallel  planes, 
and  of  indefinite  extent.  A  system  of  magnets  or  electromagnets  is 
supposed  to  exist  on  the  positive  side  of  this  sheet,  and  to  vary  m  any 
way  by  changing  its  position  or  its  intensity.  We  have  to  determine 
the  nature  of  the  currents  induced  in  the  sheet,  and  their  magnetic 
effect  at  any  point,  and  in  particular  their  reaction  on  the  electro- 
magnetic system  which  gave  rise  to  them.  The  induced  currents  are 
due,  partly  to  the  direct  action  of  the  external  system,  and  partly  to 
their  mutual  inductive  action ;  so  that  the  problem  appears,  at  first 
sight,  somewhat  difficult.  . 
2.  The  result  of  the  investigation,  however,  may  be  presented  in 
a  remarkably  simple  form,  by  the  aid  of  the  principle  of  images, 
which  was  first  applied  to  problems  in  electricity  and  hydrokinetics  by 
SirW.  Thomson.  The  essential  part  of  this  principle  is,  that  we  con- 
ceive the  state  of  things  on  the  positive  side  of  a  certain  closed  or 
infinite  surface  (which  is  really  caused  by  actions  having  their  seat 
on  that  surface)  to  be  due  to  an  imaginary  system  on  the  negative 
side  of  the  surface,  which,  if  it  existed,  and  if  the  action  of  the  sur- 
face were  abolished,  would  give  rise  to  the  actual  state  of  things  m 
the  space  on  the  positive  side  of  the  surface. 
The  state  of  things  on  the  positive  side  of  the  surface  is  expressed 
by  a  mathematical  function,  which  is  different  in  form  from  that 
which  expresses  the  state  of  things  on  the  negative  side,  but  which  is 
identical  with  that  which  would  be  due  to  the  existence,  on  the  nega- 
tive side,  of  a  certain  system  which  is  called  the  Image. 
The  image,  therefore,  is  what  we  should  arrive  at  by  producing, 
as  it  were,  the  mathematical  function  as  far  as  it  will  go— just  as,  in 
optics,  the  virtual  image  is  found  by  producing  the  rays,  in  straight 
lines,  backwards  from  the  place  where  their  direction  has  been  altered 
by  reflexion  or  refraction. 
3.  The  position  of  the  image  of  a  point  in  a  plane  surface  is  found 
by  drawing  a  perpendicular  from  the  point  to  the  surface  and  pro- 
ducing it  to  an  equal  distance  on  the  other  side  of  the  surface.  If 
the  image  is  of  the  same  sign  as  the  point,  as  it  is  in  hydrokinetics 
when  the  surface  is  a  rigid  plane,  it  is  called  a  positive  image.  If  it 
is  of  the  opposite  sign,  as  in  statical  electricity,  when  the  surface  is 
a  conductor,  it  is  called  a  negative  image.  The  image  of  a  conduct- 
ing circuit  is  reckoned  positive  when  the  electric  current  flows  in 
the  corresponding  directions  through  corresponding  parts  of  the  ob- 
ject and  the  image.  The  image  is  reckoned  negative  when  the  direc- 
tion of  the  current  is  reversed. 
In  the  case  of  the  plane  conducting  sheet,  the  imaginary  system 
on  the  negative  side  of  the  sheet  is  not  the  simple  image,  positive 
or  negative,  of  the  real  magnet  or  electromagnet  on  the  positive 
