572  Mr.  R.  Hargreaves  on  some  Ellipsoidal 
the  particular  case  with  //.  =  0  being 
+  my  +  nz)2s  day  _  usa 
^3/a  ~  (2s  +  l)A'+1'' 
1    C(hr 
(U) 
ua    and   ua  on    the    right    hand  have   (asyz)    for  variables, 
uA  and  u    on  the  left  have  (Im  ii).     We  shall  deduce  (4)  from 
f  da 
4tJ   (t*A  +  A*V^       VKQi) 
(5) 
an  integral  obtained  indirectly  in  the   previous  paper,  and 
first  give  an  independent  proof  of  (5).     The  volume  integral 
tl  j  dwdydz 
f      z2 
where  the   range    of  integration    is    the    volume    contained 
x2      i/2        ~2 
by  the    ellipsoid     —  _i_^_+_=l     is    well    known.      Put 
a-     /3Z      y 
(a,y,z)=r(l,m,7i),  and  use  a  polar  element  of  volume 
r2drd(o,  where  dco  is  an  element  of  area, on  a  unit  sphere  ; 
then  integrate  with  regard  to  r,  the  boundary  value  being 
l=>&& +  -+■$)■     The  result  is 
47T 
or 
If         I /I2       m2       n2Y3<2 
-  {  dcojial2  +  bm2  +  era2)3/2  =  —L= . 
-^  J  \/abc 
If  the  ellipsoid  is  referred  to  general  axes,  then  na  (I,  m,  n} 
takes  the  place  of  al9  +  bm?  -f  en9, ',  and  a  be  must  be  replaced  by 
Aa.  If  again  for  a  we  write  A  +  /up,  for  a',  Al  +  fipr,  then 
Aa  is  replaced  by  A  (//,),  a  determinant  with  the  elements. 
A  +  Atp, ...  and  this  is  (5). 
The  step  to  (4)  is  made  by  using  an  operator 
sWi+^iHo. 
Since  d       n         .      ,9      d  n 
j£  ua  {1,  'nh  n)  =  I2,  ^7  uA  =  2mn, 
we  have  0  .  uA  (I,  m,  n)  —  (Ix  +  my  +  nz) 2. 
