16 ON THE METHOD OF LEAST SQUARES. 



quantity which is to be made a maximum for a is propor- 



tional to 



<t> fo - ) <t> (*,-) * (*-) 



Equating to zero the differential of this with respect to a, 

 we find 



It 



as the equation for determining a in a;. Let = ^, then it be- 



comes 



2j ^ (a? - a) = 0. 



Now we have assumed that the most probable value of a is given 



by the equation 



2(a>-a) = 0: 



and it is impossible to make these equations generally coincident, 

 without assuming that 



= me, m being any constant ; 



6'e 



^ 

 <P e 



6'e 

 hence 



Now as the error e is necessarily included in the limits 

 oo + co , we must have 



r 0e^e= 



2?r 

 or if we adopt the usual notation, and replace m by 2A 2 , 



Consequently, we are thus led to adopt one particular law of 

 probability of error as alone congruent with the rule of the 

 arithmetical mean. 



But, in fact, we are perfectly sure that in different classes of 



