Closest Fit to Systems of Points in Space. 569 



AR* and e 4 /R' 2 . Hence for the semi-axes or max.-min. 

 values : 



AR*=*7R'* ...... (xxi.) 



Equations (xviii.) and (xx.) will now give the same values for 

 the ratios of the #'s and of the a/% or for any axis : 



^/x^ = x 2 /xj = .v 3 /x 3 ' = . . . =m q lxj . . (xxii.) 

 But (xxii.) combined with (xxi.) gives us : 



^"Va^' * 2 ~^W> '•'>*«= -J2W (xxm - } 



In other words, if we define points given by (xxiii.) to be 

 corresponding points,— i. e., if corresponding points lie on the 

 same line at distances inversely as each other from the origin, — 

 then the ends of the principal axes of the two ellipsoids are 

 corresponding points. Thus the principal axes of the correla- 

 tion ellipsoid coincide with those of the ellipsoid of residuals 

 in direction, and a minimum axis of the one is a maximum 

 axis of the other and vice versa. We therefore conclude : 



(i.) That the best fitting plane to a system of points is 

 perpendicular to the least axis of the correlation ellipsoid, and 

 that if 2 R m j n be the length of this axis the mean square 



residual = </ A x R min where A is the well-known deter- 

 minant of the correlation coefficients. 



(ii.) The best-fitting straight line to a system of points 

 coincides in direction with the maximum axis of the correla- 

 tion ellipsoid, and the mean square residual 



= */<r a * 1 + o J, * a + <r** J + . . . +<7%-A. R 2 max., . (xxiv.) 

 where 2 R max. is the length of the maximum axis. 



We have thus the properties of the best-fitting plane and 

 line in terms of the correlation ellipsoid, which is the one 

 generally adopted for variation problems. At the same time 

 our investigation shows us that the q directions of independent 

 variation and the standard-deviations of the independent 

 variables may be found from the ellipsoid of residuals, which 

 will usually be a process involving much simpler arithmetic. 

 (7) Numerical Illustrations. 



Case (i.). Find the best fitting straight line to the following 

 system of points supposed of equal weight : 



lV = y=5-9 0=4-4 y = 3-7 



.?• = -9 y = 5-I <v = 5"2 y~^'^> 



ar=l-8 ,y=4'4 ■ a?=6'l y=2*8 



,v = 2'6 y=4-6 a?=6-5 # = 2'4 



[x=3'3 y=3'5 a?=7-4 #=1-5 



