198  Mr.  J.  W.  L.  Glaisher  on  certain  portions  of 
to  the  limit  as  n  increases,  since  (2)  has  e~hd    for  its  limit  only 
02  1 
when  —  is  of  the  order  -,  and  the  limits  of  the  integration  are 
n  n 
from  0=0  to  0  =  oo  .     The  quantity  to  be  considered  is 
"!^(1_#0*  +  A04 )ndQ9    ...     (3) 
f 
;0 
which  we  may  write 
1 
"■^(l-AV+...)-W 
+J7^{J>)«-£r}>  •  •  ;';(*) 
where  a  is  a  finite  quantity :  the  first  of  these  integrals  is  equal 
to  l   sin— 7r-e~h262  -*->  which  differs  from  Erfc  [..-  ,    )  by 
Jo        Vn  0  \2hVnJ    J 
We  can  see  in  a  general  way  that  this  integral  must  be  much 
smaller  than  the  previous  one,  owing  to  the  rapid  decrease  of  the 
exponential  factor.     Also,  as  a  is  at  our  disposal,  subject  only  to 
the  condition  that   -7-  must  be  of  the  order  -7-,  we  can  by 
v  n  vn  J 
taking  it  large  render  the  latter  integral  very  small  indeed. 
With  regard  to  the  second  integral  in  (4),  since  6  is  always 
greater  than  —7-,  I      <j>(e)  cos  ^-j—de  must  differ  in  defect  from 
(f>(e)de(  =  l)   by  a  finite  quantity;  so  that  the  integral  is 
I 
less  than  ^/3n(/3<l),  which  may  be  neglected  compared  with 
I  have  been  unsuccessful  in  several  attempts  to  prove  rigo- 
rously the  perfect  legitimacy  of  replacing  (3)  by  Erfc  (  J 
when  n  is  large.  There  appears  to  be  a  real  difficulty  inherent 
in  this  portion  of  the  reasoning,  as  similar  ambiguities  present 
themselves  at  the  corresponding  points  in  Laplace's  and  Ellis's 
investigation.  There  would  be  no  difficulty  if  we  might  take  n  to 
be  an  infinity  of  a  superior  grade  to  the  infinite  limit  of  the  in- 
