Planes of Closest Fit to Systems of Points. 371 



theory of multiple correlation is now clear, for a t can be 



written — 

 A' = 



A'« 



rhere 



oo 



1 + ^_ 



°0 



CTqCTi 



ron + 







^02 + 



^0 a 2 J 



1 + 



ro» + 



CTqCTh 



1 + 



Thus this determinant can be derived from the multiple 

 correlation determinant by increasing r rf in the latter by 

 V s • ^"fj an( i by increasing the constituents of the leading- 

 term by the corresponding V s 2 , where Y s and V* are the 

 coefficients of variation of the coordinates x s and x t . If the 

 fixed point is at the mean o£ the system of given points, A' 

 becomes at once the ordinary multiple correlation deter- 

 minant. 



Putting n=l, we derive the two-dimensional case of a 

 line passing through the origin and giving closest fit (mea- 

 suring in the direction of y) to a system of points. The 

 equation of the line is easily seen to be 



B 10 S(*.y) 



(a) 



Putting n = 2, we reach the three-dimensional case of 

 a plane passing through the origin and giving closest fit 

 {measured in the direction of z) to a system of points. Its 

 equation is 



Z = 



l*a 2^02 — ItoiK 



01 J - t 22 



RjxRgg" 



R 



x + ■ 



Ri^Rm — IvnoR 



"12-^01 



02^11 



12 



R 11 R 22 " 



R12 



%-y 



S(j;y).S(yg)-S(^).S(y 2 ) Sfay) . S(^)-S(^) . S(* 2 ) 

 S(., 2 ).% 2 )-{S(^)p *+ S(*»).S(tf)-{Bta0>« 7J ' 



For values of n^>2 it is more convenient to derive the 

 coefficients direct from the determinant, and there is no 

 need to write them in full. 



(B) k — 1. — Here we have the case of a plane — in n dimen- 

 sions — passing through two fixed points and fitting most 



08) 



