Potentials,  JEolotropic  and  Isotropic.  577 
separately.     The  first  is 
s~  L  1_^r+  5     ---v+i. 
S      "3.5...2s,  +  3 
and  the  second  is 
VlAx2  4      2/T 1 gCl 
8  '  Ll.3...iV-rl      3.5....2sf  +  3 
___  _j-ir "I 
+  ■',2*  +  1.2*  +  3,..2*  +  2«'  +  lJ 
=  3AA  2.  4. ..25  +  25'       . 
8      "3.5  ...2s  +  2s'  +  r  "  V     y 
and  this  quantity  multiplied  by  /js  A  is  the  composite  term 
in  the  energy  o£  ksYs  +  ksiYsr.  The  summations  for  (16)  are 
made  by  a  series  of  steps  in  which  the  combination  formula 
C  =  C      4-    C     is  used,  and  the  difference  of  adjoining  terms 
^  r     s— 1  r       s—  1    r—  1 
taken.     The  densities  attaching  to  Us  and  Vs  are  respectively 
p(s  +  lua— sul*1}   and    sp(l—ua)s~l,  and  the    corresponding 
,      ,       ,  3pr0  3pT0       2  . 4 . . .  2s  , -. 
total     charges     = —        -  and  -77— . •     tney 
6        25  +  125  +  3  ^       3. 5. ..25  +  1  J 
follow  from  those  of  yfrs1  or  may  be  got  by  independent  work. 
The  energy  formula  for  Us  may  be  got  directly,  viz.  for  the 
composite  term 
E(5,50=J/vIWt 
=  J?  (7+iui  s'ut1)  *r  .  ^  <(1~jJ^  which  by  (13) 
=  3py0r(/+1)  r  __    1      =  _         1         •> 
4       Lv         ;  C25  +  I  2s  +  25'  +  3      25  +  3  2a  +  2«/  +  5J 
~*  1  27+I25T27+1 ""  27+3  2«  +  2?  +  3  /  J 
=  3p2r0(/>0/(25  +  25'  +  1)(L>5  +  25'  +  3)(2«  +  2/  +  5), 
in  agreement  with  (15).  Moreover,  the  energy  of  i/r5  may  be 
calculated  from  that  of  Us  by  treating  yfrs  as  Us  +  Us+i  +  .  .  . 
ad  inf.,  and  making  a  double  summation  of  the  formula  just 
obtained.  Either  U,?  or  Ys  for  5  =  0  gives  the  case  of  uniform 
volume  distribution  e  =  prQ,  and  energy  =  €2^a/10r0. 
