Multiplying  (334)  and  (335),


(:-"!-!:.[


c-"^^^+^'^kdAdh


„     2fl+A^)L  Jo

1  r^  ds


-,-/,2(l+A2)(_2/),)(l+A2)(i/i


tan- 'A


Therefore,


i:


0        4

Vtt


e-"dt  =  -


(334)
(335)

(336)

(337)
(338)
(339)
(340)

(341)


LAW  OF  ERROR  215

186.  Expansion  of  Law  of  Error  in  Series.  The  Law  of  Error
may  be  expressed  in  the  form  of  series  for  convenience  in  evaluating
it  for  various  values  of  A.^

Using  the  quantity,  t  =  hA,  as  an  auxiliary  variable,  we  can
state  (333)  as  follows :

«^(A)rfA  =  -^e-"'^'hdA  =  ^e-"dt  (342)

Vtt  Vtt

which  is  the  probability  of  an  error  between  A  and  A+c/A.  The
probability  of  the  occurrence  of  an  error  less  than  A  will  be  that
of  an  error  between  the  limits,  —A  and  +A,  that  is,  since  t  =  hA,

+A       r+A  1      r+t  9     r+t

p      =  <p(A)dA  =  -^\      e-'\lt  =  ^\      e-'\H      (343)

-A     J- A  VirJ-t  Vtt  J

=;7?('-3Ti+5^-4  ■  ■  •  )  (^^'


or,


for  use  with  small  values  of  t,

for  use  with  large  values  of  t.
187.  Tables  of  the  Law  of  Error.  From  the  above  formulas,
tables  have  been  computed  with  the  argument  t,  giving  the  prob-
al)ility  of  an  ci-ror  less  than  A,  in  a  given  set  of  observations.
Tabk^  I,  in  Api)endix  F,  has  been  formed  in  this  manner.  To  use
such  a  table,  the  mean  square  error  of  a  single  observation,  (e),
is  computed  fi'om  the  residuals  of  tlie  mean.  Then  i  is  obtained
from  the  assumed  error,  A,  by  means  of  the  relation,

t  =  }iA^     ^-^  (34())

eV2
siiic(>,  from  (209),

e\  2

Finally,  with  /  as  an  argmnenl,  the  tabular  ])r()t)al)ilit >'  is  obtaiiUMJ.
'  Sc(^  \\'njiht  and  HayfonK  Art.  2iy  Craiidall,  Art.  114;  dp  ( 'hauvc-nct,
Vol.  I,  Art.  ii:;.
