254  Mr.  A.  Russell  on  the 
solution.  It  is  not  easy  to  give  a  simple  proof  o£  (21),  but 
we  have  found  above  by  elementary  considerations  the  first 
two  terms.  If  we  expand  the  expression  (20)  in  powers  of 
icja  we  get 
J~    ^32 'a      256V      256"a3      *     '      *    K'  } 
The  difference  between  the  values  of  /  given  by  (21)  and 
(20)  when  x\a  is  small  is  roughly  the  hundredth  part  of  x/a, 
and  as  /  is  greater  than  unity  it  will  be  seen  that  the  per- 
centage error  made  by  using  (22)  instead  of  (21)  is  small. 
In  Table  III.  below  the  values  of  the  column  headed /have 
been  found  directly  from  the  series-formula  (15).  In  calcu- 
lating this  column  I  have  to  acknowledge  the  help  I  received 
from  four  of  my  pupils,  Messrs.  Hewitt,  Hoggett,  Bitter,  and 
Taylor.  In  the  second  column  the  numbers  are  calculated  by 
(21),  and  in  the  third  column  by  (20). 
I  am  indebted  to  Mr.  Arthur  Berry,  of  King's  College 
Cambridge,  for  showing  me  how  the  direct  calculation  can  be 
greatly  simplified.     The  formula  (15)  may  be  written 
_#       (1+y)2      fK&__2   g        fn  \ 
for* 
Kifes  (p*-w 
—  2* 
2tt       71  +  ^-1' 
We  have  used  this  theorem  to  check  several  of  our  results. 
For  instance,  when  x\a  is  0*5,  q  is  also  05  by  formula  (1). 
Also 
t   y/2/cK/ir = 2q\(l  +  tq»2+») 
i 
=  2ql(l  +  qz  +  q«  +  g"  +  q*°  +   ..  .) 
=  2*(i  +  0-25  +  0-015625 
+  0-000214+  .  .  .) 
=  21(1-26587). 
Therefore  JcK      -a   1  ...  ,„nrTV1 
-—  =2~5  (l-2bo8/)2 
=  1-1331. 
Also         «  QZn 
v       ,  *      ltq  =0-07001. 
id  +  f-1)2 
We  thus  find  that  /=  1*1 72 6,  when  #/a=0'5. 
*  A.  Enneper,  Elliptische  Functionen,  p.  179. 
t  A.  G.  Greeuhill,  '  Elliptic  Functions,'  p.  303. 
