414  Prof.  Challis  on  the  Hydrodynamical 
the  principle  of  continuity  is  concerned,  the  two  motions  might 
coexist.  It  may  also  be  readily  shown  that  this  coexistence  is 
compatible  with  the  equation  of  constancy  of  mass.  For  if 
Uyj  vv  wL  and  u2J  t>2,  w2  represent  respectively  the  components 
of  the  velocity  for  the  two  motions,  and  if  u=ux  +  u2,v=vl  +  v2, 
w=w}+w2,  we  have,  as  already  proved, 
dux      dvx      dwl  _n     du2      dvq      dwq_„ 
doo       dy        dz        '    dx       dy        dz  ""   ' 
and,  by  adding  the  two  equations, 
du      dv      dw  _  n 
dx      dy      dz  ~~ 
Having  by  this  result  proved  that  the  component  motion  fulfils 
the  condition  of  constancy  of  mass,  we  have,  further,  to  inquire 
whether  udx  +  vdy  +  wdz  can  be  made  integrable  by  a  factor; 
and  if  so,  by  what  factor.     For  this  purpose  we  have 
V?/  Yx  „       s 
u  =  ul  +  u2— -,}     v  =  Vi  +  v2= —t     w  =  wlJtwl2=j{x,y)3 
which  values  of  u,  v,  and  w  evidently  show  that  udx  +  vdy  +  wdz 
is  not  an  exact  differential.  In  order  that  it  may  be  made  inte- 
grable by  a  factor,  the  following  equation  of  condition,  as  is 
known,  must  be  satisfied  : 
/dv  ^dw\        /dw      du\         /du      di>\_n 
\dz      dy  J       \dx       dz)         \dy      dx) 
Putting,  for  shortness,/ for  f{x,  y),  and  for  V  its  value  F(r),the 
result  of  substituting  the  values  of  u,  v,  w  will  be  found  to  be 
ydf      xdf  rF(r). 
fdy+fdx"    "*"  F(r) 
This  equation  makes  it  evident  that  the  equation  of  condition 
cannot  be  satisfied  unless  /  be  a  function  of  r.  Putting  there- 
fore f(r)  for  fj  we  have 
rf'jr)  rV(r) 
f(r)    ~i+  P(r)' 
and  by  integration, 
F(r)  ~b\ {) 
b  being  an  arbitrary  constant.  This  result  establishes  a  relation 
between  f(r)  and  F(r),  by  taking  account  of  which  it  will  be  seen 
that 
udx  +  vdy -f  wdz=r¥(r)  (      ^~~^- V  -y  )  • 
