Theory  of  Magnetism.  411 
?)  =  <),       .       .       .       .      {€) 
dt  rw    ^y9  +  ^ 
which  takes  account  only  of  space,  time,  and  motion.     Since 
u=X  ~3  v=\  -t~,  iv =X  -^->  it  follows  that  the  left-hand  side 
dx  dy  dz 
of  this  equation  is  the  complete  differential  coefficient  of  -^  with 
respect  to  the  coordinates  and  the  time  ;  so  that  (  -~  )  =  0,  and 
by  integration  yjr  =  C,  an  arbitrary  quantity  not  containing  t. 
Hence,  since  ty  does  not  change  with  the  time,  the  equation 
^  —  C  =  0  shows  that  each  surface  of  displacement  maintains  an 
invariable  position.  Now  there  are  only  two  ways  in  which  this 
condition  can  be  fulfilled  when  the  forms  of  the  elements  are  also 
invariable ;  either  the  motion  is  in  straight  lines  perpendicular 
to  a  fixed  plane,  or  in  circles  about  a  fixed  axis. 
18.  First,  let  the  motion  be  in  directions  perpendicular  to  a 
plane,  which  we  will  suppose  to  be  the  plane  of  xy,  and  let  the 
velocity  along  any  line  the  coordinates  of  which  are  x  and  y  be 
f(x}  y) .  Then  we  have  u  =  0,  v  —  0,  w  =f(%,  y) ;  so  that 
udx  +  vdy  +  wdz  becomes  f{x,  y)dz,  which  is  not  integrable  per  se, 
but  plainly  may  be  made  integrable  by  the  factor  j r.     Then 
(cfyf)  =  — -dz  =  dz-j  and  by  integrating,  i|r  =  ^  +  ^(^).     But 
j\x)  y) 
it  is  shown  above  that  i/r  is  independent  of  t ;  so  that  ^(7)=0. 
Hence,  since  -\|r  is  equal  to  a  constant  C,  z—  C  =  0  is  the  general 
equation  of  the  surfaces  of  displacement,  which,  accordingly, 
are  planes  perpendicular  to  the  axis  of  z.  Also  the  motion  will 
be  the  same  at  all  points  of  a  given  filament  of  the  fluid  parallel 
to  the  axis  of  z ;  but,  since  w=f(x,  y),  it  may  be  supposed  to 
vary  from  one  filament  to  another.  The  proposition  is  thus 
proved  for  this  case,  the  result  having  been  obtained  by  means 
of  a  factor. 
19.  Next  let  the  motion  be  in  circles  about  the  axis  of  z. 
Then,  V  being  the  velocity  at  the  distance  r  from  the  axis,  we 
have  at  the  point  xyz 
Vy  Yx 
u  = -,     v  =  — ,     iv  =  0, 
r  r 
V 
and  udx  +  vdy -\- wdz  = (ydx—xdy),  which  is  not  an  exact 
differential.     It  is  evident  that  the  factor  ^    will  make  it  such, 
Vr 
