26  Prof.  Challis  on  the  Mathematical  Theory 
tion,  the  expression  for  <f>  which  is  appropriate  to  the  present 
problem.  This  might,  I  think,  be  effected  by  means  of  Laplace's 
coefficients ;  but  in  the  treatment  of  the  problem  of  oceanic  tides 
I  was  led  to  the  required  expression  for  $  by  a  particular  pro- 
cess, which  is  given  at  length  in  the  articles  in  the  Philosophical 
Magazine  already  cited.  The  solution  in  that  instance  suggested 
the  form  of  expression  which  I  now  assume,  namely 
r<f)  =f(r)  cos2  X  sin  2  (0  -  /it) . 
This  value  of  r$  will  be  found  to  satisfy  the  foregoing  equation, 
provided  the  form  of/(r)  be  determined  by  integrating  theequation 
Sf- («-¥)£-»• 
Putting  for  shortness'   sake  q  for/l  —      ^  j9  the   integra- 
tion gives 
/(r)  =  r*(Cr?+  CV"T), 
C  and  C  being  arbitrary  constants.     Consequently 
<f>  =  r-2{Cr%+  C'r"1)  cos2\  sin  2{0-fit). 
I  take  occasion  here  to  say  that  this  value  of  <j>,  which  will 
subsequently  appear  to  be  indispensable  for  accounting  theore- 
tically for  the  phenomena  of  atmospheric  tides,  has  not,  as  far  as 
I  am  aware,  been  obtained  before. 
The  fraction  — ~-  is  so  exceedingly  small  that  without  sensible 
Zoa* 
error  q  =  l.     Hence,  for  the  present  purpose, 
<£=(Cr2  +  CV-3)cos9\sin2(0-/^)-  '.  .  .  (4) 
Now  let  u!  be  the  velocity  of  the  particle  in  the  direction  of  r 
produced,  v1  its  velocity  in  the  direction  westward  from  the  me- 
ridian passing  through  its  position,  and  w'  the  velocity  northward 
along  that  meridian,  so  that 
(d(j>)  =  v!dr  +  r  cos  \iJd0  +  rw'd\, 
an  exact  differential,  because  udx  +  vdy  +  wdz  is  an  exact  differen- 
tial, whatever  be  the  directions  of  the  axes  of  rectangular  coordi- 
nates.    Hence,  from  the  equation  (4), 
^=  ^  =  (20-3CV-4)  cos2  A,  sin  2(0-^), 
t/  =  — L-  ^=2(Cr  +  C'r-4)cos\cos2(0-/*O, 
r  cos  X  du 
m/=  1  #  =      ,Cr  +  c,r_4)  sm  2\  sin  2(0-/*/). 
r  d\  v  ' 
