﻿IN THE METHOD OF LEAST SQUARES. 183 



iu whicli m is the element derived from observation, and e the unknown error of the 

 equation, to be solved by the method of least squares, giving for x the value x, with its 

 probable error, f, obtained from a comparison between the observed and the computed 

 values of m, after substituting x, y, Szc. in the primitive equations. 



If .r„ be the true value of x, we may represent by x a quantity such that it is 

 an even chance whether Sg — x is comprised between the limits i -\- x and s — x. 

 The magnitude of the limit defined by x has an evident relation to the question how 

 far the simplification of the arithmetical processes may be carried Avithout detriment 

 to the results. 



For instance, the solution of the above equations may be repeated with small varia- 

 tions from the process at first applied, giving for x a new value .r, with a probable 

 error £[, difi"ermg but little from e. If we were in entire ignorance of the relative 

 amount of the probable errors £i and e, there Avould be no reason at all for giving 

 the preference to x rather than to Xi . If only the single cu-cumstance were known 

 that f, exceeded f by a given small amount, we should be equally at a loss, while 

 the value of f remained unknown, to state the relative weight of x^ compared with 

 X, and should, in fact, be again obliged to resort to the hypothesis that s and fj were 

 sensibly equal. And in general, the greater the uncertainty of s, or, in other words, 

 the larger the value of x, the less reason would there be for excluding from competition 

 with X any other determination of x, such as x^, of which the probable error Si dif- 

 fered but little from e. 



In order to employ the limit x as here proposed, its value must be known before the 

 computations have reached an advanced stage. That this is not ordinarily practicable 

 will readily appear. On the other hand, it must be left entirely to the judgment of the 

 computer to decide as to the precise manner in which x is to be applied in limiting the 

 allowable amount of difi"erence Si — s. 



Objections of a similar character apply equally to other standards which might be 

 proposed for the same object. As has before been remarked, the question must be 

 treated, if at all, upon a somewhat arbitrary basis, and we must be content with sug- 

 gestions addressed to the judgment or common sense of the computer, in cases where 

 no fixed rule is admissible. 



Viewed in this light, there will ordinarily be no difficulty in recognizing the point 

 at which there will be danger of compromising accuracy in the attempt to simplify the 

 computations, nearly enough at least for practical purposes, if we are prepared to 

 admit, at least in its general spirit and tendency, the truth of the following propo- 

 sition : — 



