Laplace's  Proof  of  the  Method  of  Least  Squares.        199 
tegral,  so  as  to. replace  (1 f-  . . .  J  by  e~h262  throughout  the 
whole  extent  of  the  integration  with  regard  to  6  ;  but  practically 
n  is  merely  a  large  finite  number.  It  is  probable  that  a  perfectly 
general  and  rigorous  mathematical  demonstration  of  the  law  of 
facility  cannot  be  given  on  Laplace's  principles ;  but,  at  all  events, 
the  precise  nature  of  the  assumptions  necessary  for  its  truth 
might  be  investigated  with  advantage.  This  could  probably  be 
best  effected  by  careful  examination  of  one  or  two  special  cases ; 
but  in  those  that  follow  I  shall  make  no  attempt  to  examine  the 
point  just  noticed  very  carefully;  it  will  only  be  shown  that, 
admitting  it,  the  law  is  verified.  To  simplify  the  expressions, 
the  laws  of  facility  <f>v  02,  &c.  will  all  be  supposed  the  same,  and 
/Ltj,  /x2,  &c.  will  be  taken  each  equal  to  unity. 
Taking  first  the  case  discussed  by  Leslie  Ellis,  viz.  when 
<^>(6)  =  ^6>:Fe,  the  lower  sign  being  taken  when  e  is  negative,  the 
integral  (1)  becomes 
2  f*  f  f  "  e6  ,  \ »  sin  6  jn 
2  P°   •     9  fi  _!_  e*  Y*l* 
"ija  ™75V1+aO  dd 
2fao  .    e     *de     %  -, .,  /  i  \ 
=  -l      sin-7-e    s-a-  =  -7^Erfchr-7-), 
on  the  supposition  that  we  may  in  the  last  two  integrals  imagine 
the  infinite  limit  replaced  by  a  finite  quantity  when  we  please, 
or  that  we  may  imagine  n  increased  absolutely  sine  limite.  This 
investigation  is  very  much  shorter  than  Ellis's,  which  is  ob- 
tained by  expanding  the  circular  function  in  ascending  powers 
of  0*.  It  might  for  the  moment  appear  as  though,  since  posi- 
tive and  negative  errors  are  equally  likely,  the  law  of  facility 
<f>(e)  must  be  a  function  of  e2.  It  is  clear,  however,  that  such  a 
law  as  that  taken  above,  viz.  e~^e2}  is  quite  as  admissible; 
there  is  no  reason  why  the  algebraical  expression  for  the  law 
should  be  continuous. 
There  are  not  a  great  many  forms  of  (f>  for  which  $(e)  cos  ae  de 
can  be  integrated  between  the  limits  0  and  co  in  finite  terms. 
The  following  two  cases,  however,  are  instances  of  such  forms : 
*  Camb.  Trans,  vol.  viii.  p.  213.     In  the  integral  there  discussed  cos  6 
takes  the  place  of  —q-  ;  the  reasoning,  however,  is  not  affected  thereby. 
Ellis  assumes  n  to  be  so  great  that  it  is  greater  than  the  number  of  terms 
in  the  series  for  cos  6  ;  i.e.  he  takes  it  to  be  an  absolute  infinity. 
