Theory  of  Magnetism.  413 
and  v  =  0,  does  not  exclude  finite  values  of  -7-  and  7— ,  and  con- 
'  dx  dy 
sequently  it  is  possible  that  w  may  vary  from  one  line  of  motion 
to  a  contiguous  one.  Thus  it  has  been  shown  that  the  supposed 
motion  in  parallel  lines  satisfies  all  the  general  equations  (e)} 
(A  and  fo). 
2.2.  Proceeding,  now,  to  the  case  of  rotatory  motion  about 
the  axis  of  z,  it  will  be  found,  on  substituting  in  the  equation  (/) 
for  u,  r,  w  the  respective  values — 3  — -,  0,  that  the  result  is 
_ydv  +  w_dv=0_ 
r  dx       r     dy 
This  is  a  partial  differential  equation,  the  solution  of  which  by 
the  usual  process  is  V=F(r).  It  is  thus  proved  that  the  circular 
motion  is  a  function  of  the  distance  from  the  axis  and  of  arbi- 
trary value.  It  remains  to  ascertain  under  what  dynamical  con- 
ditions this  kind  of  motion  is  possible. 
23.  Since  w  =  0}  the  equations  to  be  used  for  this  purpose  are 
dp  .   du        du        du  __  n  „ 
dx      dt         dx        dy~   ' 
dp      du        dv         dv n 
dy      dt        dx        dy  ~~ 
By  substituting  in  these  equations  the  foregoing  values  of  u  and 
v,  it  will  be  found  that 
,n  V2  _  dV  .  '  . y 
(dp)  =  —  dr  —  r-rrd.  tan-1  -■ 
v  ri       r  dt  x 
In  order  that  the  right-hand  side  of  this  equation  may  be  an 
dN 
exact  differential,  we  must  have  -^-  =  0;   so  that  V  is  a  function 
of  r  without  containing  t,  and  the  motion  is  thus  shown  to  be 
steady.     Hence  also 
dr    "   r  ' 
that  is,  the  centrifugal  force  is  counteracted  by  variation  of  pres- 
sure with  the  distance.  Since  the  right-hand  side  of  this  equa- 
tion is  necessarily  positive,  the  pressure  p  continually  increases 
with  the  distance. 
24.  Suppose,  in  consequence  of  what  has  now  been  proved, 
that  for  the  case  of  motion  in  parallel  straight  lines  we  have 
y^{  =  Cv  and  for  the  circular  motion  -v|r2  =  C2,  and  that  the  sum 
of  the  two  equations  gives  -\Jr  =  C.  Since  this  composite  equation 
is  of  the  same  form  as  the  components,  it  follows  that,  so  far  as 
