570  Mr.  R.  Hargreaves  on  some  Ellipsoidal 
Thus  the  surface-discontinuity  or  density  is  the  same  for 
all  the  functions  ^r  and  is  that  of  yjr0  ;  while  potentials 
Us  and  Ys  containing  differences  of  powers  of  ua  have  only 
a  volume-density.     A  second  differentiation  gives 
d2yfrs   _  p     /   d2\         J   d\d\\        sp     ^ua  d\ 
dxdy  4zs/J\dxdi/      2 J  dx  dy  J      ±\/3~dx    dy 
When  Ve^s  is  formed,  the  sum  of  terms  such  as  those  in 
the  first  bracket  vanishes,  and  the  total  outside  the  sign  of 
integration  is  —sp/  \/J.  Under  the  sign  of  integration  the 
bracket  gives 
2i4-1  2  {pa  +  2p'a!)—4:(s  —  1)  u~2  ua, 
or 
2usa-1j/J-±(s-l)usa-%a  quoting  M.  (76)  and  (S6). 
Hence  the  integral 
Externally  the  lower  limit  is  variable,  and  the  two  terms  are 
cancelled  in  virtue  of  ua  =  l  ;  thus  ^/eyjrs  =  0.  Internally  the 
lower  limit  is  constant  and  =  0,  and  there  are  no  terms  outside 
the  integral  sign;  since  A,  =  0  makes  J  =  l,  ua=ua,  the  in- 
tegral is  spUa~L,  that  is  Vl^rs  =  spu~1  and  the  volume-density 
is  -sjdmJ"1.     The  surface-density  a 
per 
-5T  being  a  perpendicular  from  the  centre  on  a  tangent  plane 
and  I  m  n  direction-cosines  of  a  normal. 
Since  —~  =  -^  4-   ^ ,  and  — /-?  and  -f^.  are  external 
dx         dx         0$  dx  dx 
