248  Mr.  A.  Russell  on  the 
sphere  Y.     Let 
A1An+i  =  u/l+i  and  BBn  =  u'n, 
then  the  above  equations  may  be  written 
(d-O)V  =  a2  =  u^d—u^) 
(d  —  u^iL-l  =  a2  =  u3(d—u2') 
In  general,  we  have 
(d  —  Un^^n-l    =  CI2  =  Un(d  —  lLn-i), 
and  thus 
Un{d  —  cfi/(d  —  Un-l)}  =  «2, 
iinun-i  —  {(d2 — a2)jd}un  —  (a9/d)un-i  =  —a2. 
This  form  of  difference  equation  is  well  known  *  and  is 
readily  solved  by  assuming  that  un  =  vn+i/vn+(d2  —  a2)d. 
Making  this  assumption  we  find  that 
^+1  +  {(d2-2a2)/J}^+(aV^>»-i  =  0, 
a  linear  difference  equation  with  constant  coefficients.    Hence 
solving  in  the  ordinary  way  f  we  get 
vn  =  Aan(a/d-  q)n  +  Ban(a/d  —  ljq)71, 
where  A  and  B  are  constants  and 
2q  =  d/a-  \/d2-4:a*la,       .      .      .      .    (1)  J 
and  2/q  =  d/a  +  <s/d*-4a?/a,       ....     (2) 
so  that              i  /     ,           j/  /o\ 
l/g  +  q  =  dla, (3) 
and  1/9-9=  \ltf-±a*\a (4) 
Now  when  n  is  unity  Wi  =  0,  and  thus 
v2/Vl  =  -  (d2-a2)/d  =  -a{l  +  q2  +  q±)l\q(l  +  q*)\. 
Substituting  for  v2  and  v1  their  values,  in  terms  of  A  and  B, 
in  this  equation  we  find  that  B  =  —Aq2.      We  thus  find  on 
*  Boole's  'Finite  Differences,'  3rd  ed.  p.  233. 
f  Boole's  'Finite  Differences/  chap.  xi. 
j  I  have  called  this  expression  q  so  as  to  introduce  elliptic  function 
notation.  It  is  a  pure  number  and  has  nothing  to  do  with  an  electric 
charge. 
