[ 367 ] 



XLIII. On Restricted Lines and Planes of Closest Fit to 

 Systems of Points in any Number of Dimensions. By 

 E. 0. Snow, M.A* 



Statement of the Problem. 



I. FJHHE theory of the lines and planes of closest fit 

 JL to systems of points when no restriction is placed 

 upon those lines and planes has been developed by Prof. 

 Pearson in various papers f and is of frequent application. 

 The connexion of these lines and planes with the formulas of 

 the theory of multiple correlation is indicated in those papers. 

 If the criterion of " closest fit " is that the sum of the squares 

 of the deviations from the line or plane measured in the 

 direction of the " dependent " variable is to be a minimum, 

 the equation of the line or plane is identical with the corre- 

 sponding multiple correlation formula. If the sum of the 

 squares of the deviations measured at right angles to the line or 

 plane is a minimum (and this, from the purely geometrical 

 point of view, is the more satisfactory criterion), the result is 

 not of such a simple form, but the determinant from which the 

 mean square residual is obtained is similar to the multiple 

 correlation determinant. 



While working on certain vital statistics, it was desired 

 to obtain a formula connecting the "dependent" variable 

 with the "independent" variables when the values of all 

 the variables were known at the beginning and end of a 

 certain range, and the correlation between " dependent " 

 and " independent " variables at all intermediate points was 

 a maximum. Thus, if x Q denote the " dependent J ' variable, 

 and #!, x 2 ) ' - • x n the "independent " ones, we require to 

 make the correlation of x with # 1? x 2 > . . . x n a maximum, 

 with the condition that when ar l9 «%, . . . x n take up the 

 values pn, jt? 21 , .. .p n i, pi2,p 2 2- • -Pn2 respectively, x Q is to 

 take the values j? i an d jd 02 . 



A similar problem occurs in certain branches of Physics, 

 especially in connexion with solutions and alloys. A 

 property — e. g., the freezing-point — of a pure substance 

 may be definitely known, and it is required to investigate 

 the behaviour of that property as certain amounts of some 

 other substance or substances are added. Fixed conditions 

 will be imposed upon the law which is to be investigated by 

 the known properties of the pure substance. The law, then, 



* Communicated by Prof. Karl Pearson, F.E.S. 



t See Phil. Mag. Nov. 1901, pp. 559 et seqq. ; Phil. Trans, vol. elxxxvii, 

 A. pp. 301 et seqq. 



