Closest Fit to Systems of Points in Space. 565 



plane perpendicular to the best-fitting line is perpendicular 

 to the least axis of the ellipsoid of residuals. Thus we find : 

 That the line which fits best a system of n points inq-fold space 

 passes through the centroid of the system and coincides in 

 direction with the least axis of the ellipsoid of residuals. 



The mean square residual (which measures o£ course the 

 closeness of the fit) is given by 



\Jo\ 



+ a\ + ...+ <r 2 Xq — H2> • • (xi v.) 



where R is the least radius of the ellipsoid of residuals. 

 The direction-cosines of the line can be found from (ix.) by 

 giving £ 2 the least value among the roots of (x.) . 



Clearly the plane of best fit passes through the line of best 

 fit, and is further perpendicular to the greatest radius, the 

 maximum axis of the ellipsoid of residuals. 



(5) While the geometry of lines and planes of best fit is 

 thus seen to be very simple from the standpoint of inertia 

 ellipsoids, — particularly from the consideration of the surface 

 which, for the theory of errors, I have termed the ellipsoid of 

 residuals, — they most frequently occur, perhaps, in the case 

 of correlated variations or errors, and it is thus of interest to 

 consider them in relation to the ellipses and ellipsoids which 

 arise as " contours" in correlation surfaces. 



Now take the case of two variables x and y only, the 

 type-ellipse of the contours of the correlation surface is, 

 when referred to its centroid as origin : 



x*_ f__ 2r xy xy _ 

 °~ x °" y o~x °"y 

 Compare this with the ellipse of residuals 



<T 2 x X** 4 (T°-y y 12 + 2<Tx <Ty r X y rftf = 6 4 . 



Clearly if we take a f =y, y'= — #, and e 4 = cr 2 ro- 2 y the ellipse of 

 residuals becomes the correlation type-ellipse. Further, 

 a/ 2 + 3/' 2 = # 2 + ?/ 2 , or the two ellipses have equal rays, but they 

 are at right-angles to each other. Thus the best-fitting 

 straight line for the system of points coincides in direction 

 with the major axis of the correlation ellipse, and the mean 

 square residual for this line 



product of standard deviations 

 ~~ semi-major axis of correlation ellipse " 



Phil. Mag. S. 6. Vol. 2. No. 11. Nov, 1901. 2 P 



