578  Mr.  R.  Hargreaves  on  some  Ellipsoidal 
§  5.  We  now  give  corresponding  results  for  ^.s.  Using 
the  integrals,  (10  b)  for  the  first  step,  and  (12  b)  for  the 
second, 
\AusXax  +  cfy  +  b'z)dT^ 
USa(a.V  +  ry'!/  +  ;3'z)d\ 
=  I  LiCax  +  c'y  +  b^dr  fAr^  +  ^H^)^1^ 
=  3  r   Aq/^ft) 
(25  +  3X27+27+5)  J  2ttwi,wa3/2 
3L0  " 
-(25  +  3)(25-r25,  +  5j (17> 
where  L0  as  in  M.  (135)  stands  for 
™  *dX   _    C  AJhlco 
1 
J 
This  and  the  corresponding  surface  integral  may  be  written 
^(a,+cV+^)^T=___w|_)    .   (m) 
and 
(<«  +  CV  +  i'%!WS=j|^) (17  c) 
The  surface-density  due  to  %s>  is  ^j     (a*  +  c'y  +  6'~),  and  the 
volume-density  — s'/m*'~1(a#  +  e/y  +  &/2)  ;  thus  applying  (17  b 
and  c)  the  composite  term  in  energy  attaching  to  indices 
s  and  s'  is 
Bf,/w-5&k  Fi — 1  = 3p2r°L°        (is) 
•Ejw;-8(25  +  3j[-L      2*  +  2«''+3J       8(2^  + 2/ +  3)*     UC; 
If  functions  with  yr x  +  j3y -\- exf z  and  ft'  x  +  a'y  +  yz  are  also 
under  consideration,  the  three  functions  may  be  distinguished 
as  %*(#),  Xs(y)>  %«(-)•  I11  the  composite  term  for  %s(y)  and 
%s-  (y),  M0  appears  for  L0  above,  but  if  a  composite  term  for 
%*(#)  and  ^  (j/)  is  in  question  N0'  appears,  viz., 
ivj  /_  f  AJmdco  _  C*  y'd\ 
For  the  density  of  ^  iy)  nas  the  factor  c'#  +  by  +  ar^  ;  and  if 
this  is  taken  with  %«(#)  a  factor  wi  is  introduced  in  place  of  I 
in  the  second  step  towards  (17 J,  where  (12  b)  is  quoted  ; 
while  the  contrary  arrangement,  density  of  %s(a')  with  potential 
