﻿IN THE .METHOD OF LEAST SQUARES. 187 



one adopted expressly on account of its good qualities, though without bearing any in- 

 tended resemblance to the method of least squares in its characteristic features, will 

 diminish the difference ,?' — c?', to a value either less in absolute amount than s, or very 

 nearly equal to it; so that (.r — x^), {y — 3/,), &c, may be sufficiently well expressed 

 by the series (2) in the same sense that t may be. The equations 



ax-\-ly -\- -j- „i = e, 



«•»! + * i/i + + m, = 0, 



in which e is the value of e obtained by substituthig x\ y, Sec. in the primitive equa- 

 tions, give at once 



(5 c.) a {x — X,) -\- h (y — y^) -{- + (,« _ ,«,) = g, 



The solution of which by the method of least squares gives (a; — d\), (y — y{) , 



and thence 



a; = a-, + (a; — a;,), 



y = yi + (.y — m)^ 



Therefore the application of that method may be confined to the equations (5 c) alone. 



But by what has already been said, (,v — o^J, (y — yi) , and e, are quantities 



sufficiently well expressed by the series (2), from which we readily infer that there can 

 be no appreciable advantage in girag to (m — ?«i) or to the products a (.r — .r,), 

 b(y — yi), &c., or to the factors a, b, &c., any higher exactness of expression. Hence 

 the proposition (2) may be extended to every term and factor of the equations (5 c), 

 and therefore to all the nimierical processes peculiar to, and characteristic of, the meth- 

 od of least squares. 



This extreme application cannot, however, be recommended even on the ground of 

 convenience or simplicity ; on the contrary, the indiscriminate use of the fractional 

 terms of the series would often be highly inconvenient; and a form of solution like 

 that just indicated would not always be deskable. 



In considering the different sources of error from which b and Si acquke their value, 

 it will be convenient to compare the increase given to a by the introduction of small 

 intentional inaccuracies, in consequence of which s becomes f,, either with s itself or 

 with s — T), by means of the following relations derived from (-4) : — 



(6.) 



f, E 



<Kn"- ^<m 



