412  Prof.  Challis  on  the  Hijdrodynamical 
and  we  shall  thus  have 
v 
Hence  by  integrating,  ^=tan~1-,  no  arbitrary  function  of  t 
oc 
being  added,  because  it  has  already  been  shown  that  ^  is  equal 
to  a  constant  C  which  is  independent  of  the  time.     Consequently 
C=tan-1-,  or  ?/=^tanC,  C  being  an  arbitrary  arc.      This 
os 
general  equation  of  the  surfaces  of  displacement  indicates  that 
the  motion  is  in  circles  about  the  axis  of  z.  This  result  having 
been  arrived  at  by  means  of  a  factor,  the  proposition  that 
udx  +  vdy  +  v;dz  is  integrable  by  a  factor  for  this  kind  of  motion 
is  thereby  demonstrated. 
20.  It  is  now  to  be  observed  that  although  the  general  equa- 
tion (e)  is  satisfied  by  the  two  supposed  kinds  of  motion,  the 
possibility  of  such  motions  is  not  proved  till  the  other  general 
equations  have  been  taken  into  account.  Yet,  according  to  the 
essential  principles  of  applied  calculation,  the  circumstance  that 
that  general  equation  has  been  satisfied  cannot  be  without  sig- 
nificance ;  and  it  is  on  this  account  necessary  to  inquire  whether 
and  under  what  conditions  the  other  general  equations  are  satis- 
fied by  the  same  motions. 
21.  Taking,  first,  the  motion  in  parallel  straight  lines,  since 
w=0,  v  =  0,  and  w=f(x,y))  it  is  evident  that  the  general  equa- 
tion of  constancy  of  mass, 
du      dv      dw     „ 
is  at  once  satisfied,  and  it  only  remains  to  take  account  of  the 
dynamical  equations 
2+eH  g+©=°>  l+S)=°-  •  « 
By  substituting  the  values  of  u}  v,  and  w  in  these  equations 
there  will  result 
^=0     *=0     ^-0- 
dz     V)    dy3    dz~V' 
whence  it  follows  that  (dp)—0}  and  p  is  constant.  Since  the 
last  of  the  three  equations  is  equivalent  to 
dp      dw        dw         dw         dw 
it  may  be  remarked  that  the  foregoing  reasoning,  since  u  =  0 
