Closest Fit to Systems of Points in Space. 571 



Let us find the best-fitting plane, treating these as four 

 points in three-dimensioned space. We have at once 



* =3, #=21, 7=209-5 



<r*=l, a y = 5, <r*= 50-0275 



rzycryaz— 182*5, rxz<rxo-z= — 30*5, rxydxo-y—O. 

 Thus the ellipsoid of residuals is : 



^ 2 + 25/ + 2502-75^+365^-61^ = e 4 . 



The equations to find the direction-cosines are : 



(2 + 25V + o.Z 8 -61.Z 8 = 0, 

 O.Z,+(50 + 29y 8 + 365/ 8 =0 



-61^ + 365^+ (5005'5 + 2^V 3 = 0. 



Whence writing 2 — =^ (71 = number of points = 4) the cubic 

 for % is: n 



C= x 3 + 5057-5x 2 + 123,440x + 48,050 = 0. 



We want the least root : 



X =0,C=+; X =--5,C=-j X =-100, C=+; X =-», 



C=— . Thus the required root lies between and —'5. 

 It is easily found to be 



Xl =- '395,660. 



Thus — — ='197830, and the mean square residual 

 n 



= \/ - ^='4448. 

 V n 



We easily deduce : 



h 1% [3 



38-02187"" 7-35823 ~ 1" 



Thus the best-fitting plane is : 



38*02187 (07-3) -7-35823 (</- 21) +2-209-5 = 0, 

 or: 



2 + 38-02187*- 7-35823^-169-03778 = 0. . (xxv.) 



If we find the values for z for given x and y, say those of the 

 four points, which are 



211-7, 135-7, 283-3, 207-3, 



we should not be impressed by the goodness of the fit. But 



