﻿IN THE METHOD OF LEAST SQUARES. 191 



Probable value of (x — a;,) u. Probable value of (v — y.) u. 

 / 15 \ > L' ^ — g := g ■ — ^ ^—gz=g 



Probable value of (xo — ^) /^ Probable value of (i/o — y) ^o 



Aud from (12) the general expression 



(16-) 1"= g. 



Giving to r/ the vahie (3) 



we shall have in the present case 

 and by (6) 





£ ^ 23 



1 



f. 



^ 1250 



In other words, by using the form of solution (11.) in the place of a rigorous appli- 

 cation of the method of least squares, the probable errors of the concluded results will 

 not be increased by one one-thousandth part, — a difference entii'ely too small to be 

 sensible. The two processes, as far as regards accuracy, therefore, may be consid- 

 ered as perfectly identical. On the other hand, the advantages of simplicity and 

 convenience are altogether in fa^•or of the second, in which all the operations of mul- 

 tiplication and division required in the construction of the final equations are reduced 

 to their simplest arithmetical forms. 



The necessity of distinguishing between the probable error of a\, that is, the proba- 

 ble value of (a'o — .^i), and the difference between a;, and x, or (a', — a.'), deserves par- 

 ticular attention here. While a\ wUl often differ very much from a\ this fact taken by 

 itself by no means indicates that the chances that .v^ is the true value are not sensibly 

 as good as that x is. The discrepancy really proves that the original observations upon 

 which the discussions have been based are so faulty, that very little confidence can be 

 placed in either result, or in any other that can be deduced from the same data. 



To give completeness to the investigation, we will compare the processes by which 

 the probable errors of the values of the iudeterminates in the two solutions (I.) and (II.) 

 are obtauied. 



For the system (I.), let cV, y^ ~, &c. be eliminated in succession from the equations 

 (8), in the following manner. 



Multiply the final equation for x by — and subtract it from the final equation for y, 



and again by — and subtract it from the final equation for .:, forming the new equa- 



P- 



F 



tions, in which x does not enter : — 



