Theory  of  Magnetism,  403 
of  these  three  physical  forces,  I  shall  now  endeavour  to  give  as 
exact  a  proof  of  it  as  may  be  possible.  The  mathematical  rea- 
soning relating  thereto  under  the  head  of  the  Theory  of  Electric 
Force  in  pages  545-547  of  my  work  '  On  the  Principles  of 
Physics '  is  incomplete  and  inaccurate. 
4.  Suppose  a  current  of  the  sether  to  traverse  a  substance  con- 
sisting of  atoms  so  arranged  that  their  number  in  a  given  space 
increases  regularly,  but  by  very  small  gradations,  in  a  given  di- 
rection, and  conceive  the  whole  of  the  space  occupied  by  atoms 
to  be  very  small  compared  with  the  intervening  space.  Also,  the 
motion  being  (at  first)  assumed  to  be  steady,  let  V  be  the  ave- 
rage velocity,  and  p  the  average  density  of  the  eether  at  a  certain 
position  A,  and  let  B  be  another  position  distant  by  &*  from  A 
in  the  direction  in  which  the  atomic  density  increases.  Then  if 
1)  be  the  proportion  of  the  space  occupied  by  atoms  to  the  whole 
of  the  given  space,  or,  as  it  may  be  called,  the  contraction  of 
channel,  the  quantity  of  fluid  which  passes  a  unit  of  area  at  A 
in  the  unit  of  time  is  Vp(l  — D).  But  it  has  been  proved  (see 
Phil.  Mag.  for  June  1864,  p.  458)  that  when  a  stream  is  inci- 
dent on  a  small  sphere,  the  mean  flow  in  the  original  direction 
is  not  altered  by  the  disturbance  of  the  lines  of  motion  caused 
by  the  reaction  of  the  sphere;  and  the  same  is  the  case  if  there 
be  many  such  spheres,  provided  the  contraction  of  channel  they 
produce  is  very  small.  Hence,  supposing  the  initial  generation 
of  the  stream  to  have  been  such  that  the  same  quantity  of  fluid 
was  made  to  pass  each  element  of  a  given  transverse  section  in  a 
given  small  interval,  according  to  the  above  reasoning  this  con- 
dition will  be  fulfilled  at  auy  subsequent  epoch  at  each  transverse 
section,  notwithstanding  the  presence  of  the  atoms;  so  that  the 
quantity  of  fluid  which  passes  through  an  elementary  space  at 
B  will  be  the  same  in  the  same  interval  as  that  which  passes 
through  an  equal  space  at  A.     Consequently 
S,Vp(l-D) 
hz 
Hence,  passing  from  small  differences  to  differentials,  the  rea- 
soning being  independent  of  the  magnitude  of  6>,  we  have 
dp             dV              ^V  M 
£5~"(1-D)«fe  +  Ydz"K) W 
Now,  the  motion  being  stead}T,  the  fluid  unlimited  in  extent,  and 
no  extraneous  force  acting,  if  p0  be  the  value  of  p  where  V  =  0, 
the  equation  p  =  p0e>  >2«2  is  applicable  at  all  points,  even  when  the 
effect  of  the  reaction  of  the  atoms  is  taken  into  account.     Con- 
2D2 
