410  Prof.  Challis  on  the  Hydrodynamical 
certain  effects  of  motions  of  the  air — for  instance,  the  sounds  of 
recognizable  pitch  resulting  from  the  mutual  collision  of  aerial 
streams,  or  from  the  diversion  given  to  such  streams  by  encoun- 
tering solid  obstacles,  are  explainable  on  the  same  principles. 
15.  I  take  occasion  here  to  remark  that  the  generation  and 
propagation  at  the  surface  of  water  of  a  series  of  circular  waves 
in  forms  which  appear  to  be  independent  of  the  mode  of  dis- 
turbance, or  shape  of  the  disturbing  body,  are,  I  think,  referable 
to  dynamical  reasons  analogous  to  those  adduced  above.  Also 
the  series  of  small  waves  which  are  seen  to  precede  a  cylindrical 
rod  when  it  is  held  vertically  and  moved  horizontally  through 
water  in  which  it  is  partly  dipped,  may  be  similarly  accounted 
for.  (These  "  ripples/-'  together  with  the  broader  waves  which 
follow  the  rod  under  the  same  circumstances,  are  described  and 
discussed  by  Professor  W.  Thomson  in  an  article  in  the  Philo- 
sophical Magazine  for  November  1871.)  Supposing  the  fluid 
to  be  one  of  perfect  fluidity,  the  foregoing  argument,  which  is 
based  on  that  supposition,  leads  to  the  conclusion  that  the  gene- 
ration of  the  ripples  may  be  ascribed  to  the  obstacle  opposed  to 
the  motion  of  the  rod  by  the  inert  mass  of  fluid  in  front,  and 
that  the  waves  behind  are  broader  than  those  before  by  reason 
of  the  reluctance  with  which  the  mass  behind,  on  account  of  its 
inertia,  follows  the  rod. 
16.  The  motions  which  have  been  thus  far  considered  are  all 
such  that  each  element  of  the  fluid  is  at  each  instant  changing 
its  form,  and  the  lines  of.  motion  are  normals  to  continuous  sur- 
faces, so  that  udx  +  vdy +  wdz  is  always  and  everywhere  an  exact 
differential.  This  may  be  true  even  supposing  the  motion  to  be 
in  directions  perpendicular  to  a  given  plane,  because,  as  I  have 
indicated  above,  the  rectilinear  motion  may  be  composite,  in  which 
case,  the  change  of  form  of  the  fluid  elements  takes  place  with 
respect  to  each  of  the  component  motions.  There  are,  however, 
cases  of  the  motion  of  a  fluid  in  which  each  element  maintains 
always  the  same  form,  either  because  the  whole  mass  moves  or 
rotates  as  if  it  were  solid,  or  consists  of  an  unlimited  number  of 
parts  which  individually  so  move.  Such  motions  are  distinguished 
by  the  analytical  circumstance  that  for  them  udx  +  vdy  +  wdz  is 
integrable  by  a  factor.  To  prove  this  is  the  object  of  the  fol- 
lowing argument,  in  which,  for  the  sake  of  brevity,  the  fluid  is 
supposed  to  be  incompressible. 
17.  For  proving  a  proposition  of  this  kind  it  is  necessary  to 
employ  the  general  equations  of  hydrodynamics,  in  order  that 
the  reasoning  may  depend  on  the  fundamental  principles  which 
these  equations  express.  I  shall  therefore  begin  by  drawing 
an  inference  from  that  which  I  call  the  equation  of  continuity, 
namely : — 
