200     On  Laplace's  Proof  of  the  Method  of  Least  Squares, 
p_cosA*_         # 1_  5. 
«.  -x 
Q2a^-e    2a 
f°°  e~c2*2  cos  5#  j 
a«  +  *«      dX 
=  ^!e*w^(r«Brf(ac_A)+<-»Erf(flc+  ^J}*  (6) 
Let  us  take,  therefore,  <ft(e)  =  —   a       _M>  and  the  resultiug 
7»    &    ~i  e 
integral  is 
2  rx/       3        ysinfl    . 
^    (  ^    .*)  -r* 
,,0     \e-^74-e   2ui  ' 
Similarly,  if,  in  accordance  with  (6),  we  take  as  our  law 
a  e'0'2^ 
^  =  2\/ireaie2'Ev{ac  ~aT^?i 
the  resulting  integral 
Poissont  has  demonstrated  that,  if  6(e)  =  - =.  the  law  of 
it  1  -f  e"-' 
facility  is  not  of  the  form  e_7i"e2.  This  arises  from  the  disconti- 
nuity in  the  value  of  l      —r — it  which  =  ~  eTa.    It  can  be  easily 
Jo      1"fe  * 
shown  that  the  same   discontinuity  occurs  if  6(e)  be  propor- 
tional to  -^ -A  kc.     This  virtually  includes  any  rational  alge- 
braical law  1  for  <f)(e)  must  not  be  infinite,  either  when  e  is  finite 
or  infinite;   so  that  the  most  general  expression  is 
<£(e)=2  ^A^+5;_?_  +  &c. . 
T  w  e*  +  a*         e4  +  a 
thus  the  law  of  facility  would  not  be  proportional  to  e~h:*':  if 
the  facilities  of  the  individual  errors  were  rational  algebraical 
functions. 
"With  reference  to  the  proofs  of  the  law  of  facility  that  have 
* 
Phil.  Mag.  tec.  cit.  p.  298. 
t    Connaissance  des  TejiwsAS'2 
