Laplace's  Proof  of  the  Method  of  Least  Squares.       197 
The  reasoning  that  occurs  near  the  conclusion  of  the  above 
investigation  may  also  be  exhibited  in  another  form,  which  is 
perhaps  clearer. 
Resuming  equation  (1),  we  may  write  it 
2fxsin0r 
^Jo    -^-{l-/^2  +  ...}...{l-/^2+...}^. 
& 
Put  6=     /——,  and  this  becomes 
s/Vi] 
■    -    -  —  -    -      -...  ^     l-         ...us 
do 
2  r»oo  sm 
f^H{,-g+...}...{l-g+...}, 
2p  .     e        do 
=  -  I     sin— 7=^e-*—i 
as  before.  The  legitimacy  of  the  neglect  of  the  terms  beyond 
02  is  best  seen  by  taking  the  simpler  case  of  01  =  ^)2=&c., 
^  =  ^2=  &c. ;  the  expression  is  then  of  the  form 
/        hW^      A04  V 
\}^ir+r<~h   •  •  •  •  (2) 
the  limit  of  which  clearly  is  e~h'262.     It  is  to  be  remarked  that 
fx       sin  Q 
Q-n     -     d0  being  infinite  in  no  way  prejudices  the 
reasoning,  as  we  are  concerned  with  the  series 
c 
*fl^(l_A02  +  B<94...)^ 
as  a  whole ;  and  the  function  in  brackets  must  always  be  less 
than  unity,  since  it  is  the  product  of  n  factors  of  the  form 
fee 
<£(e,)  cosae^;  and  this,  </>,(e,)  being  always  positive,  is  less 
than  1     $(6i)dei  (that  is,  than  unity). 
It  is  worth  while  to  examine  rather  more  carefully  the  approach 
