582  Mr.  R.  Hargreaves  on  some  Ellipsoidal 
together  with  (26) .     The  first  with  (26)  gives  2©  +  C  {2p - 1) 
=  2p—  1,  and  the  right  hand  of  the  q  +  l\th  equation 
=  2ay  +  C{2p-f2q-l)  =  2p  +  2g-l^^^i, 
which  i£  the  stated  minimum  (23)  is  correct 
The  group  of  equations  is  then 
+  ... 
2p  +  2q  +  l       2p  +  2q±3      '-'2p  +  2q  +  2n—l 
=     //H^n  t<2P  +  2n  + 1) - 2gr]     .     (27) 
where  ^  has  in  succession  the  values  0?  1,  2  ..n ;  if  these  are 
satisfied  the  original  conditions  are  satisfied  and  co  has  the 
value  in  (23). 
If  Pj^(^)  is  used  for  the  rth  differential  coefficient  of  the 
zonal  harmonic  Pn{v<),  the  solution  of  (27)  is  comprised  in  the 
statement  of  the  following  as  an  identity,  viz.  : 
A'1  +  ^-T+^         1.3  1.3.5 
2    2^+1   ,        42£+12p  +  3         /f62p  +  l  2^  +  3  2^  +  5 
(28) 
gprWi 
n(2/i  +  2»  +  l)1.3...2p-l 
The  numerical  coefficients  on  the  left  hand  are  cleared  by  an 
integration   I    /<iyu,,  followed  by  p  —  1  integrations 
•/=  ]  oA*/aA*J 
I 
and  the  result  is 
,^-  +  ,2/^  +  -^^-  =  _^^_rijjj; ;    (29) 
for  p  =  l,.^iA*+>^»  +  ...      =P^+1/n(2n  +  3),       ...     (296) 
and  for    p  =  2,  ^/*8  +  ^  +  •••     =  ^{^1+2-^+2) /n(2n  + 5).  (29c) 
The  left-hand    member   of  (27)   is    got   from   the   left-hand 
member  of  (29)  by  multiplying  by  yLt2?+1  and  integrating  from 
