﻿182 ON THE USE OF EQUIVALENT FACTORS 



aloue. No one can pretend that this is a process susceptible of strict accuracy ; yet an 

 error here is as fatal as if we had disregarded any other of the precepts of the method. 



This step being an arbitrary one, although one of fundamental importance, we may 

 properly appeal to it as a precedent for the modification of others suggested by con- 

 siderations of convenience, though they may not, like this, be justified on the plea of 

 actual necessity. In this Yiew of the subject, we find support for the modification 

 suf^o-ested by Gauss, in the passage we have quoted above. Each of the complicated 

 factors which it is there proposed to simplify is itself a product of two other factors, 

 one of which is the weight of the equation under treatment ; if one of these, that 

 is, the number representing the weight, is erroneous, the product is of course errone- 

 ous, with whatever accuracy the other is expressed. 



Attain, as a matter of convenience, it is usual to express the conditional equations 

 proposed for solution in a linear form, by reducing the indeterminates to small quanti- 

 ties and neglecting the terms multiplied by their second and higher powers, and to con- 

 struct from them normal equations, as they are called, previously to applying the 

 method of least squares. Both of these may be practices perfectly allowable under the 

 circumstances, but since they are almost always theoretically incorrect, their admis- 

 sion is a vu-tual relinquishment of aU pretensions to a rigorous course of computation, 

 and cannot be compensated for by any subsequent refinements. 



We will now proceed to examine the limits of accuracy appropriate to the arithmeti- 

 cal operations required in the combination of conditional equations by the method of 

 least squares, and afterwards to develop in detail some proposed modifications of that 

 method, haviu"- for their object the reduction to its minimum value of the amount of 

 labor requisite for its successful application. 



It is scarcely necessary to remark, that the subject is plainly one Avhich is in its 

 nature somewhat vague and insusceptible of rigorous treatment, though it is at the 

 same time interesting from its practical bearings. If no very precise or definite rules 

 for re<^ulatino- the degree of numerical exactness suited to the discussion of any given 

 problem can be arrived at, it may still be of service to point out the principles which 

 ought to guide the computer in the choice of such limits as shall perfectly meet all 

 reasonable requirements of accuracy, without imposmg upon him the unprofitable labor 

 of multiplying the extent and difficulties of calculation, to no useful purpose, and 

 without the remotest prospect of sensibly unproving the real value of the results. 



Let us suppose a series of equations, 



a X + J y + -\- VI = e, 



a X -\-b' y -\- -^ni = e', 



