Laplace's  Proof  of  the  Method  of  Least  Squares.         195 
deduction  of  the  law  of  facility  of  2^e,  when  n  is  made  very 
large,  is  also  presented  in  a  form  somewhat  different  from  that  of 
either  Laplace  or  Ellis;  and  the  general  formula  is  verified  in 
several  instances  by  assuming  special  laws  for  the  individual 
errors,  so  as  to  render  the  n  integrations  capable  of  accurate  per- 
formance. The  principle  made  use  of,  which  is  due  to  Lejeune 
Dirichlet,  depends  on  the  discontinuity  of  the  integral 
2foosin0  .    . 
—  \      —a—  cos  <yu  aU 
c  0 
(which  =0  if  7>1,  but  =1  if  7  lies  between  0  and  1),  and  may 
be  stated  as  follows  : — Suppose  the  value  of 
^...${xv...xn)dxl...dxn 
is  required  for  all  values  of  x\  . . .  xn,  subject  to  the  condition  that 
yfr(xlt  . . .  xn)  lies  between  +1;  then  we  can  replace  the  multiple 
integral  by 
Vo    J-ooJ-oo  * 
cos  {^r{xl}  . .  ,xn)9}d6  dx1 . , .  dxn, 
in  which  the  limits  are  independent  of  the  variables. 
In  the  case  to  be  considered  we  require 
If-  -  •  *<«.) ;  •  *  *-(0*i--  •  *  *W 
subject  to  the  condition 
Ahei  +  A*a€a '  •  *  +  Pit**  >  —  /  and  <  /, 
whence  the  multiple  integral  becomes 
cos 
{o(ftt-;+w)}M,  ...den. 
Assuming  the  equal  probability  of  positive  and  negative  errors 
so  that  (pi(ei)  =<£,( — e.),  then 
f 
^.(e,-)  sin^y-^e^O, 
whence  we  have 
2 
