Closest Fit to Systems of Points in Space. 



567 



(6) We turn to the correlation type-" ellipsoid " for q 

 variables. It is * : 



*i s 



#2 



k 22~2~ 



cr X2 



A„^-+A 22 ^f ...A 



0~ xi 



** , 9A x l x 2 



qq -o r ^^*12 ~~~ 



* y <J 2 x q <Txi<Tx 2 



+ 



_j_9A flg-i&'q _i 



• (xv.; 



where A n , A 22 , A 12 . . . A 9 _ l9 , A qq are the minors correspond- 

 ing to the constituents marked by the same subscripts of the 

 determinant : 



1 



l 



r 23 



r iq 

 r 2 q 



rq 2 



r Q3 



(xvi.), 



Now let us find the directions and magnitudes of the 

 principal axes of this ellipsoid. We must make 



a maximum. Or if Q be an indeterminate multiplier, we 

 have : 



(An + Q^J 



-^+A 12 -^ +A 13 



o-x, 



<Tx., 



*> + ...+A 1? ^=0, 



<Tx 



ex, 



k l? 



0~x l 



+ (A„ + QaVf s - +A„^-+ ...+A h ^. =0 , 



d.'g 



<7.r 



**3 



'*« 



Aw _^L + A 2? ^- + A 3? ^- +. . .+ (A w + Qo«, 9 ) A =0 (xvii.) 



**i 



<Tx n 



&x« 



<r Xq 



Multiply the last q — 1 of these equations by r 12 , r 13 , . . . r Y q 

 respectively and add them to the first, then we know that : 



A u +r u A u -fr ls Ai3+ ... + r lq A lq =&, 



and if u be not 1 : 



A 1 „+r 1 oA 2tt +r 1 sA 3tt + . .. +r iq A qu = 0. 



* Phil. Trans, vol. clxxxvii. A, p. 302. 



2 P2 



