

368 Mr. E. C. Snow on Restricted Lines and 



has to give the best fit to the observations made of the 

 property as definite quantities of the other substances are 

 added. Two examples of such cases are given from the 

 figures of certain alloys (§7 below). 



The idea is capable of generalization, and the theory 

 for the general case will be investigated. Looked at from 

 the point of view of correlation, we shall require to assume 

 a linear law connecting x with x u x 2 , — x n , and shall 

 make the sum of tho squares of the deviations of the actual 

 observations from this linear law measured in the direction of 

 x a minimum, making use of the exact conditions which are 

 imposed on the law. This will be first investigated. But a 

 better geometrical fit to the observations will be obtained 

 by measuring the deviations perpendicular to the "plane" 

 given by the linear law, and this will be worked out sub- 

 sequently. 



Analytical Investigation, 



First Method. 



2. Let there be n "independent" variables and (&+1) 

 conditions connecting them with the "dependent" variable. 

 (£ + 1) is necessarily less than n. Measuring from one of 

 these fixed conditions, we can assume our law is 



x = a 1 x 1 + a 2 x 2 + . ... -r-flntfn, . . „ . (1) 

 with the k conditions 



Poi = fli^ii + a 2 p 21 4- .... -f a n p n i, 



P02 = aiP\2 + a2l>22+ +«»i?7i2, 



(2) 



Pok = ^lpVc + a 2 p2k + • • • - + anpnk- 



Then we want to make 



V= &{x — a lt v l — a 2 x 2 — .... —a n oc n y, 



the sum of the square of the deviations in the direction of ^ ? 

 a minimum, subject to the above conditions. 

 Hence we must have 



= S(a' — a x xi — a 2 x 2 — .... — a n xn) (x x . da 1 -f x 2 . da 2 



+ .... +x n .da n ), (3) 

 with the conditions 



=p u , da x +p 2 * .da 2 + , . , . -f- p ng . dan. (s = 1, 2, . . . . k.) (4) 



