19G  Mr.  J.  W.  L.  Glaisher  on  certain  portions  of 
Now 
putting  k2  for  1     0e-(e.)e2Jet-;  the  first  term  is  unity,  since 
Jo 
Too 
1     <j>(et)dei=l.    Thus  (1)  becomes 
J  —  CO 
2l  f  °°  g2iog(i-/^02+..osin_^^_  2_r°°£-s^-2g2-Afl4-B0«-...sin  fl^ 
27.2 
^  being  written  for  ^~.     Now,  n  being  very  large,  2^#2  is  of 
the  order  n0'2  =  an62,  say;  similarly  A,  B,  &c.  are  of  the  same 
order;  so  that  the  integral  takes  the  form 
•7T  I     6  0 
Since  n  is  very  large,  the  exponential  is  finite  only  when  6  is 
very  small  and  of  the  order  n— ?,  so  as  to  make  nfl2  finite ;  when 
this  is  the  case,  the  other  terms  are  of  the  orders  62,  04,  &c, 
and  may  be  neglected.  It  is  to  be  observed  that  if  6  is  of  the 
order  n~ *,  so  as  to  make  n6*  finite,  the  first  term  is  of  the  order 
6~2,  and  the  value  of  the  exponential  is  infinitesimal.  We  may 
therefore  neglect  all  the  terms  except  the  first,  so  that  the  in- 
tegral becomes 
2  f00   -rfsinfl  J/x        2     _  e       1 
—  I     e         -x-dd=z  -7-  Erfc  - — -= 
=  4-Erfc 
which  is  the  well-known  result.     The  integral  made  use  of,  viz. 
I     e        dx  =  viT  \       e      du  =  virhric-n    ,  t 
Jo  oo  Jo  2^« 
is  obtained  at  once  by  the  integration  of 
A     e  ax  cos  tote  ==  — -j-  e~  u 
Jo  2*/« 
with  regard  to  b  *. 
*  See  Phil.  Mag.  vol.  xlii.  pp.  298  and  421  (October  and  December  18/1). 
