Theory  of  Magnetism.  405 
tions  made  in  1848-1857,  and  in  1858-1863.  In  the  volume  for 
1867,  p.  ccxiv,  he  says,  "We  are  justified  in  stating  that  there 
is  no  certain  evidence  foi'xlnnual  Inequality/''  The  same  result 
is  arrived  at  with  respect  to  the  Horizontal  Force.  The  annual 
inequalities  deduced  from  the  theory  in  (  The  Principles  of  Phy- 
sics '  (pp.  657-660)  were  obtained  by  employing  the. argument 
which  is  shown  above  to  be  untenable. 
It  is  true,  however,  that  General  Sir  E.  Sabine  has  deduced 
very  small  annual  inequalities  of  Dip,  Total  Force,  and  Declina- 
tion by  discussions  of  observations  taken  at  Kew,  Toronto,  Ho- 
barton,  St.  Helena,  and  the  Cape  of  Good  Hope  (see  Phil.  Trans, 
for  1863,  p.  307).  But  it  is  possible  that  these  may  be  due  to 
a  cause,  distinct  from  the  eartlr's  motions,  which  is  adverted  to 
in  art.  36  of  the  communication  on  magnetic  force  in  the  Philo- 
sophical Magazine  for  February  1861,  and  will  be  more  fully 
treated  of  in  the  sequel  of  the  present  communication. 
7.  It  remains,  therefore,  to  determine  by  what  means  the 
streams  to  which  the  theory  ascribes  the  magnetism  of  a  steel 
magnet  are  generated,  the  magnet  being  either  a  straight  bar  or 
in  the  form  of  a  horseshoe.  This  question  I  shall  endeavour  to 
answer  by  taking  account  of  the  theories  of  atomic  repulsion  and 
molecular  attraction  proposed  in  the  Numbers  of  the  Philoso- 
phical Magazine  for  March  and  November  1859  and  February 
1860,  and  m  < The  Principles  of  Physics'  (pp.  459-464).  For 
this  purpose  the  following  general  hydrodynamical  theorem, 
which,  as  far  as  I  am  aware,  has  not  hitherto  been  recognized, 
will  be  made  use  of : — Whenever  the  lines  of  motion  in  a  given 
fluid  element  are  normals  to  a  continuous  surface,  so  that  the 
element  is  changing  form  by  reason  of  the  motion,  the  function 
udx  +  vdy  +  wdz  is  an  exact  differential.  In  proof  of  this  theo- 
rem it  seems  sufficient  to  say  that  the  change  of  form  of  a  given 
element  in  consequence  of  convergency  or  divergency  of  the 
lines  of  motion  is  a  distinctive  property  of  a  fluid,  whereby  its 
motion  is  separated  from  that  of  a  solid,  and  that  the  integra- 
bility  of  that  differential  function  is  the  sole  and  necessary  ana- 
lytical expression  of  this  property. 
8.  This  being  understood,  we  have,  as  is  known,  for  a  fluid, 
defined  by  the  relation  between  the  pressure  and  density  ex- 
pressed by  p  =  a'2p)  the  general  equation 
a™  Nap.  log  p  +  J  +  ^+/W  =  0, 
where  a1  is  put  for  tea  (k  being  a  known  numerical  factor,  the 
theoretical  determination  of  which  I  have  discussed  in  previous 
communications),  and  (d<j>)  =udx+vdy-\-wdz.  No  extraneous 
force  being  supposed  to  act,  this  equation  is  applicable  to  all 
