﻿184 ON THE USE OF EQUIVALENT FACTORS 



The application of the method of least squares to the discussion of observations of physi- 

 cal phenonena, with the exception of a few special cases of rare occurrence, requires the 

 (2.) use of such numbers only, in the arithmetical jyrocesses j)eculiar to it and characteristic of 



the method, as may he designated by one of the numerals 0, 1, 2 9, or of the fractions 



-i-, — —, or by a product of one of these numbers by an integral power o/" 10. 



An idea may be formed of the amount of the intentional errors occasioned by these 

 substitutions, by noticing that if by N is represented any number whatsoever, and by 

 N' a number chosen from the proposed series which most nearly coincides with N, we 

 shall have 



JV— JV 1 

 ,„ ^ The maximum value of ~ — = - nearly. 



rr,, , , , , „ N— N' 1 

 The probable value of ~ — <[ — . 



Before proceeding to a detailed investigation of the consequences of the changes pro- 

 posed, it will be useful to point out the degree of insecurity attaching to the values 

 which must ordinarily be adopted to represent the probable error of x ; the different 

 sources which may be supposed to contribute to the increase of s ; and their relative 

 importance in connection with the question of the comparative accuracy of the two 

 results X and Xi. 



s may be referred to the combined influence of two mutually independent errors 

 7) and T)', 7) being the probable Aaluc of x^ — x which would result from the errors of 

 observation alone, supposing the theory of the method of least squares and its applica- 

 tion to the data to be rigorously exact, and 77' the probable amount of error in x having 

 its origin in errors necessarily committed in the discussion of the observed data, sup- 

 posing the mode of discussion, although the best, practicable, to fall short of strict 

 conformity with the theory. 17, represents the value of ri when the same data have 

 been reduced, by a process made intentionally still less exact, to a small extent, both 

 in its theory and in its arithmetic, than that Avhich gives the error 17'. 7][ will bear 

 to x\ a relation similar to that which if bears to x. v cannot be completely eliminated, 

 so long as the errors of obser^•ation remain unknown, by any treatment, and the same 

 may be said of 17' ; but 77,' can always be reduced to its least limit, 77', by suitable refine- 

 ments of theory and of computation. In view of the fact that 77 and 77' must have 

 always sensible, but very uncertain values, it will be of but little consequence that 

 Tj'i should be reduced to its utmost limit without regard to the labor and inconvenience 

 which it may cost. At all events, the attempt will be ineffectual as a means of im- 

 proving the substantial accuracy of the results, as we shall presently see. 



