572 On Lines and Planes of Closest Fit. 



the small value of the mean square residual shows how close 

 to each of the points the plane really goes when we measure 

 its distance from a point not by the vertical intercept, but by 

 the perpendicular from the point on the plane. Thus the 

 vertical distance from # = 4, y = 16, £=127, to the plane is 

 8*7, but the perpendicular distance is only '1988. 



If % 2 and v 3 be the other two roots of the cubic C we easily 

 find: 



%2%3 = 121,442-65, %2 + %3 = -5057-10434. 



Thus we have the quadratic to find ^ 2 and % 3 



% 2 -r5057-10434 % + 121,442-65 = 0, 

 or: 



X 2 = -24-12895, %3 = -5032-97445. 



% 3 gives the least axis of the ellipsoid of residuals ; hence the 

 direction-cosines of this axis are given by 



n ^2 h 



-•012,125 -073,249 1* 



We have accordingly for the equation of the best-fitting 

 straight line to the four points : 



#-3 ,y-21 _ ^-209-5 



-12-125 " 73-249 ~ 1000 * * ; ^ XV1 *' 



The mean square residual for this line 2' is given by (xiv.) 



= V2528-75-2516-487225 

 = 3-5018. 



This again is remarkably small, considering how far our 

 four points are from being co-linear. 



The reader will easily prove directly that the best line 

 (xxvi.) really lies in the best plane (xxv.) . 



These two illustrations may suffice to show that the methods 

 of this paper can be easily applied to numerical problems ; 

 the labour is not largely increased if we have a considerable 

 number of points. It becomes more cumbersome if we have 

 four, five, or more variables or characters which involve the 

 determination of the least (or greatest) root (as the case may 

 be) or an equation of the fourth, fifth, or higher order. 

 Still, the coefficients being numerical and all the roots real 

 and negative, it is not very difficult to localize them. 



