J. p. ENCKE ON THE METHOD OF LEAST SQUARES. 343 



may also define the 7nean error thus : it is the error which, if it 

 alone were assumed in all the observations indifferently, would 

 give the same sum of the squares of the errors as that which actu- 

 li ally exists. According to this, the probability W, of the concur- 

 ^ I rence of m true observation errors is, generally, in any hypothe- 

 0,1 sis which can be made as to the constant h of the function <^ A, 



Wn — h^m i.? 



= — ; — e 



i m 



jj|' From this value we are now enabled to determine the most 

 '1 probable value of ^; for if the m observation errors, and conse- 

 quently also e^, have actually been found, and cannot be further 

 altered, then the maximum of this function W will depend only 

 on the value of h. The most probable value of /* will be that 

 which makes this function W a maximum. 



We may first seek this maximum by the differential calculus. 

 If we write the above expression thus : 



log. W = OT log A — 1^ m log TT — ^- m e^, 

 then the condition of the maximum is 



= J- 2m A e/; 



or 



1 = 2 /i^ s/, 

 whence 



1 

 h = 



^2^2 



We may also develope generally the magnitude W as a func- 

 tion of /i, for altered values of h. Let the value W' belong to a 

 value A + A, just as the value W belongs to h, we shall then 

 have 



log W = m log (A + A) - i mlog tt - (7* + AfniB^^ ; 

 if we write here for m log (^ + A) the expression 



m log A + m log f 1 + — j 



and develope the latter part into the known series, we have 

 log W = m log A - i m log ir — h^ m b^ 



A , A^ , A^ , A" 



— 2 m s^ h A — mzl A 



