﻿IN THE METHOD OF LEAST SQUARES. 185 



Since rj is iuclependent of V and ^Z, , we liave, assuming 7/ to be the least attainable 

 value of Vi) 



Q O I /O 091/0 



If Tj" be used to designate the probable value of ,v — ,t\ Avhicli would result from 

 small intentional deviations from that treatment of the data which is recognized to be 

 the best, we have 



/A \ 'Q /O 1 //O O o I /.-> cy o I ^O I rtn 



(4.) yji~ — r,~ Jj- r, ~, £- = 7)- + ,-, f,- = ,- _j_ ^- -j- ;, 2. 



As regards the uncertainty of s, some estimate of its extent may be obtained in the 

 following manner. 



If it is an even chance that the error of which the probable value is rj is comprised 

 somewhere between the limits 57 -|- \ and t] — X, 97 having been derived from compari- 

 sons of a given system of equations with observation, the number of individual equa- 

 tions thus compared being represented by n, and the number of unknown quantities 

 entering into them by n', X may be found from the expression* 



(5.) X = 0.477 1 . 



Any value of n — n less than 100 gives 



-^ 21 



The scale of substituted numbers (2) admits, as we have before stated, of represent- 

 ing t) within the probable amount of — 17 ; hence, for any Aalue of n — n' less than 100, 

 the series will afford numbers representing 77 with a probable error less than X. A 

 slight examination will show that a similar remark applies still more decisively to e. 



The considerations which oblige us to attribute a sensible value to 77' are too many 

 and too obvious to require to be specified in detail. It will be sufficient to cite one or 

 two which have already been alluded to. The existence of unknown constant errors in 

 the data will render the application of the method of least squares, strictly speaking, 

 inexact. From this source rf will inevitably acquire some influence. Again, the lui- 

 certainty incident to any attempt to assign to the original data their proper relative 

 weights, will haxe a similar effect. No process more loose and arbitrary can well be 

 conceived, than that by Avhich the relative precision of the elements afforded directly by 

 observation is graduated. Yet, imperfect as it is, improvement in this particular is 

 scarcely to be hoped for. Exact conformity with a theory which requires a previous 

 knowledge of the relative weight of observations is quite impossible. 



* Gauss, Zeitschrift fur Astr., B. I, Theor. Comb.,>J 40. 



