574  Mr.  R.  Hargreaves  on  some  Ellipsoidal 
and  in  particular 
§  3.  The  fundamental  integral  for  moments  of  inertia  in 
an  ellipsoid  is 
L  [(U  +  mv  +  nz)*sdT- - fM?>f»,n)y 
rJ<Jx  +  mt,  +  nz)    ^-(2s+]){2s  +  d)X     ""AT")      (11) 
The  coefficients  o£  like  power  arrangements  in  Imn  are 
equal  on  the  two  sides,  so  that  all  moments  and  products  of 
inertia  of  degree  2s  are  comprised  in  the  formula.  A  way 
of  expressing  the  results  for  individual  terms,  which  also 
applies  to  (4),  will  be  given  later  ;  but  for  our  main  purpose 
the  form  (il)  and  some  collateral  results  are  required.  In 
immediate  connexion  with  (11)  are 
-J  (^+)»y+»^'„r<fr=(2s+1)(2s+2s,+3)(^)  (iu) 
and  the  surface  integral 
lj(te+my+^)2WS=_3_gy;  .  (llc) 
where  ^  is  a  perpendicular  from  the  centre  on  a  tangent 
plane.  Write  w,y,  z=r(X,  //,,  v),  use  a  polar  element  of 
volume  r2dr  do)  and  integrate  to  the  surface  value 
l  =  U2ica(X,  ya,  v). 
For  (11) 
I  {lx  +  my  +  nz)2sdT=\\  r2s+2{l\  +  ?nn  +  nv)2s dr  dco 
for  (lib)    \{lx+my  +  nzysuSadT 
=  2. +17+3  \  R2s+2s'+3(^  +  mf*  +  ™YS <$» to  v)da> 
=  2i+WTh  JR2S+3  &+«*+"")?  <**>'> 
and  for  (11  c) 
\  {lx  +  my  +  nz)2s  vd$  =  (  R2s+3  (l\  +  m/i  +  n v)2s  da>, 
since  vr  d$  =  W  day. 
