Planes of Closest Fit to Systems of Points. 373 



When these values are substituted in the equation giving 

 the fixed condition, viz. 



p = a^p v + -\-a t p t -\- + a n p n , 



X can be found, and from this the a's can be completely 

 determined. This method will be used in an example below. 

 No other particular cases of this method need be worked 

 out in detail. We see that it is alwaj's possible to obtain a 

 plane in w-dimensional space to pass through any number 

 (less than n) of fixed points and to be such that the sum of 

 the squares of the deviations of any number of other points 

 from the plane measured in a fixed direction is a minimum. 



Second Method. 



4. We have now to investigate the equation of the 

 corresponding plane when the criterion for "closest fit" 

 is that the sum of the squares of the deviations from the 

 plane measured at rigid angles to the plane is the least 

 possible. From a purely geometrical point of view this 

 will give a closer plane to the system of points, but it 

 will not give the regression of one variable on the others. 



Let /, .... l n be the generalized direction-cosines of the 

 plane, and take one of the points through which the plane 

 has to pass as origin. 



Then the equation of the plane is 



l lXl + .... +l n x n = 0, (9) 



the total number of variables being taken as n for convenience 

 of notation. There will also be the conditions 



l x * + .... + Z n 2 = 1 (10) 



and ^11+^21 + +Jnj?nl = 0, 1 



hpn + kpm + + lnPn2 = 0, 



(11) 



kpik + kp2k + + InPnk = 0, J 



(A'+l) being the total number of fixed conditions. 

 The criterion for closest fit is 



v = s(^+ .... +i„.v K y 



to be minimum, subject to the above conditions. 

 Hence 



= S(/^! + + l n Xn)(Vldh + + XndU), 



together with 



lidlx + .... + Indln — 0, 

 Pndl^i- +p n \dln = 0, 



Plkdl L + +Pnkdln = 0. 



