﻿IN THE METHOD OF LEAST SQUARES. 181 



tial to its successful application, or to the attainment of all the advantages which its 

 employment may confer upon the discussion of any practical problem 1 



It is true that no other system can be proposed which is free from similar objections, 

 or which can be mathematically demonstrated to be exclusively the best, Avithout quali- 

 fication, and therefore the arguments above stated are of no force whatever, if employed 

 as reasons for the rejection of the method of least squares. They nevertheless greatly 

 weaken the position of those who would insist upon a strict compliance with its pre- 

 cepts, and effectually preclude all arguments of a iiurely theoretical character in support 

 of such a course. Still it is desirable that the force of any objections which may be 

 made to an attempt to modify the theoretical conditions for effecting the most favorable 

 combination of equations should be appreciated at their true value. We therefore pro- 

 pose to shoAv that the spirit of the method of least squares, rightly apprehended, in 

 reality rather invites than discountenances a liberal construction of its rules. 



Admitting that the best possible solution is attained when the sum of the squares 

 of the outstanding errors, represented by /2, is a minimum, it is evident that /2 is a 

 minimum relatively to the manner in which the original equations have been treated. 

 And since the peculiarity of the solution consists in the employment of a system of 

 factors, a, a', &c., by which the original equations are multiplied before combination, 

 the first differential coefficient of fl relatively to either of these factors, in the case of 

 the least-square solution, must have the value for each factor, 



a a 



When, therefore, the factors are varied by small amounts, S a, B a, &c., the conse- 

 quent variations of 12 developed in a series, will contain only terms multiplied by the 

 second and higher powers of S a ; or, in general terms, if loe deviate from the exact 

 (1.) precepts of the method of least squares hy small variations of the first 'order, ive shall fail 

 to satisfy its fundamental criterion by small terms of the second order only. 



Looking thus at the most elementary principle of the method, we find a warrant 

 for some degree of liberty in applying it, — a liberty which we can scarcely hesitate to 

 avail ourselves of, if we further consider the peculiar circumstances attending its actual 

 employment in the discussion of data furnished directly by observation. 



Among its first requirements is the assignment of weights to the original observations ; 

 but it is one which it is not possible to fulfil correctly, for we are provided neither with 

 a theory nor with data for the purpose. All that can be done is to accept, as indices of 

 the relative value of the different observations, certain numbers depending either proxi- 

 mately or remotely upon no other authority than the mere exercise of the judgment 



VOL. VL NEW SERIES. 24 



