﻿186 ON THE USE OF EQUIVALENT FACTORS 



At the same time, then, that the existence and influence of t] are admitted, its amount 

 is altogether uncertain, to an extent sufficient at least to make the uncertainty of i 

 which is dependent on that of 17 and V not less in proportional amount than that of 7? ; 

 consequently we shall obtain from (5) the expression 



(5«.) x> 0.477-^ 



by which to measure the uncertainty of s. If n — n' < 100, 



(5i.) ''>^'- 



1 

 21 



When « is represented by a number chosen from the series (2), the probable error of 

 the representation is, by (3), less than — £ ; in other words, it is more than an even 

 chance that this number will fall within the limits f + ^ « and £ — - £ ; and since 

 the inherent uncertainty of £ makes it more than an even chance that its actual -value is 

 outside of the limits f 4" ^ ^ ^^'^ * — 21 ^' "^ accordance with the aboA'e determination 

 of X, we conclude that £ can be represented by one of the series of numbers 0, 1, 2 



9, or of the fractions -, - -, or by a product of one of these niunbers by 



an integral power of 10, with more accuracy than we can determine its amount by 

 one hundred comparisons between the observed and the computed values of m. It 

 wovxld be easy to show, from the probable existence of constant errors alone, that an 

 indefinite increase of the number of comparisons with observation would not sensibly 

 diminish the uncertainty of £ below the amount stated. The j)roposition (2) would thus 

 be sustained, as far as relates to all expressions for probable errors and weights, since 

 they must depend upon conditions similar to those limiting the accuracy of £. 



An immediate consequence of this admission will be the extension of the proposition 

 in question, in the qualified sense, at least, in which alone it is to be understood, to all 

 other arithmetical expressions required in the application of the method of least squares, 

 since the peculiar province of the latter is restricted entirely to the solution of equations 

 of the form 

 (5 c.) a (x — j;,) -f A (y — (/,) + + {m — ?«,) = e, 



in which each separate term and factor may be defined as proposed in (2). 



To illustrate this, let us suppose for the moment that Xi has been derived from the 

 same primitive equations, but by an essentially different process from that by which 

 X has been obtained ; Xi would still be precisely equal to x, if it were not for the errors 

 e, e', &c. Any such process, not intentionally bad, must evidently lead to a determina- 

 tion of Xi differing from x by an amount of an order not higher than that of f, while 



