Theory  of  Magnetism.  415 
Hence  the  required  factor  is    ^.  w  and 
Thus  tlie  differential  equation  of  any  surface  of  displacement  is 
dz=-b  **-***, 
and  by  integration  tlie  equation  of  the  surface  becomes 
s=c~btm-l% (k) 
These  results  give  the  means  of  defining  exactly  the  character 
of  the  motion.  According  to  the  principle  of  easy  divisibility, 
■we  may  conceive  the  fluid  to  be  divided  into  an  unlimited  num- 
ber of  infinitely  thin  cylindrical  shells  having  the  axis  of  s  for 
their  common  axis;  and  if  ¥(r)  be  the  rotatory  velocity  of  a 
given  shell  at  the  distance  r  from  the  axis,  then  will  f(r),  or 
—j — ,  be  the  velocity  of  the  same  shell  parallel  to  the  axis.  Con- 
sequently the  motion  of  any  given  point  of  the  shell  will  be  in  a 
spiral.  The  form  and  position  of  the  spiral  may  be  inferred 
from  the  above  equation  of  the  surfaces  of  displacement  when 
the  values  of  the  constants  b  and  c  are  given.  The  spiral  is 
left-handed  or  right-handed  according  to  the  sign  of  b. 
25.  We  have  still  to  determine  the  dynamical  conditions  of 
this  motion.  From  what  has  been  proved,  the  components  of 
the  velocity  have  the  following  values : 
u=-M¥(r),     *=?F(r),     w=f(r)  =  ^F(r); 
and  for  determining  the  pressure,  since  u,  v,  w  are  independent 
of  z,  we  have 
dp        du  t     du      r,    dp        dv        dv      _    dp  ,     diu       div     _ 
dec        dx        dy  dy        do:        dy  dz       dx        dy 
The  last  of  these  equations  gives 
|=-*|FM/'('-)  +  fW(<-)=o. 
Hence  the  equations  for  determining^  are  the  same  as  for  simple 
Up)       ya 
rotation  about  the  axis,  and,  just  as  for  that  case,  — ~  =  — , 
or  the  centrifugal  force  is  counteracted  by  the  increment  of  the 
pre-sure  with  the  distance  from  the  axis. 
(The  foregoing  investigation  takes  account  of  all  the  cases  of 
