524 Prof. F. Y. Edgeworth on the Law 



method of least squares in the reduction of observations re- 

 lating to several variables. For let it be granted that all 

 error s-qf -observation are in general effects of numerous small 

 agencies — a proposition both a priori probable and verified 

 in the case of observations relating to one quantity by copious 

 experience; then it follows from the reasoning of this section 

 that the law of frequency for the errors of two (or more) 

 quantities is of the form J<?~ R , where R is a quantic of the 

 second degree, which can be put in the form of a sum of two 

 (or, as the case may be, more) squares, such as 



A(.*-x) 2 + B(y-y) J ; 

 and then it follows, as Merriman * and others have reasoned, 

 that the method of least squares is the solution of a maximum- 

 problem : to determine the origin x, y from which the observed 

 system of values have most probably emanated. 



Again, the extended law of frequency presents on a larger 

 scale the spectacle of order emerging from chaos which is 

 formed by the evolution of one universal law in the com- 

 pound from any particular law in the elements. There is a 

 difference, too, in quality as well as in scale. When we con- 

 sider elements involving one variable only, it is only with 

 regard to laws of frequency that the melting of discord into 

 rhythm is observed ; but in the case of elements involving 

 two or more variables, that phenomenon is observed with 

 respect to statistical laws generally. Whatever, in the ele- 

 ment, is the function connecting x with the y which on an 

 average corresponds thereto ; in the compound, generally, the 

 relation between x and y will be linear. For example, let the 

 law of frequency for each element form such a cubic surface 

 that the ordinate which on an average corresponds to any 

 assigned x is given by a parabola y = Ax* ; yet in the com- 

 pound the correlation between x and y is linear, of the form 

 y—Y=r (x— X) ; where X and Y are the average values of 

 x and y (in the compound), and r is the coefficient of correla- 

 tion which Mr. Gal ton has made familiar f. By parity in 

 the case of several variables, however complicated the laws 

 connecting the variables in the elements, the correlation be- 

 tween the composite organs is of the form 



o(*-X)+%-Y)+<jr(*-Z); 



where a, h, g, <fec. are the coefficients investigated by the 

 present writer in a former number of this Journal J. 



It is not to be supposed that this conclusion amounts to no 



* ■ Method of Least Squares/ arts. 41, 47. 

 t Proc. Roy. Soc. vol. xlv. p. 140, et seq. 

 X August 1892. 



