Ixxxvi 



THEORY OF ERRORS. 



Characteristic Errors of Interpolated Logarithms, etc. 



2. The Method of Least Squares. 



a. General statement of method. 



When the errors to which observed quantities are subject follow the law ex- 

 pressed by 



*w = 7; '-'■■•■• 



a unique method results for the computation of the most probable values of the 

 observed quantities and of quantities dependent on the observed quantities. The 

 method requires that the sum of the weighted squares of the corrections to the 

 observed quantities shall be a minimum,* subject to whatever theoretical condi- 

 tions the corrections must satisfy. These conditions are of two kinds, namely, 

 those expressing relations between the corrections only, and those expressing 

 relations between the corrections and other unknown quantities whose values are 

 disposable in determining the minimum. A familiar illustration of the first class 

 of conditions is presented by the case of a triangle each of whose angles is mea- 

 sured, the condition being that the sum of the corrections is a constant. An 

 equally familiar illustration of the second class of conditions is found in the case 

 where the sum and difference of two unknown quantities are separately observed ; 

 in this case the two unknowns are to be found along with the corrections. 



Mathematically, the general problem of least squares may be stated in two 



* Hence the term least squares. 



