for a System of Correlated Variables. 



19 



and A, is given by the deter min an tal equation of the nth 

 .degree, 



"13 



m 1 



(^-x) hi ... 



(--4 



!: 2 i 



W2i 



^31 

 M 3 



Wl 3 



1 



> = 



(B) 



I£ X be any real root of equation (B), its substitution in (A) 



determines the ratios — , — , &c. in a system of values o£ 

 iti u i 



Mi..Ai n for which Q is minimum, given E. 



Also 



Q=i2w^9 =-- }x2mtt'=\fiE, 



u 2 u 3 



and this, with the ratios — , — , &c, determines an actual 



set of values of u x ...u n which make Q minimum, given E. 

 34. Now expanding equation (B), we have 



X. n -A 1 X,»- 1 + A 2 \»- 2 -...±A ft =0;. . . (C) 



in which Ai=S — , and is the sum of the roots, A /4 is the 

 m 



determinant B when \=0, and differs from D only by the 



factor , and is also the product of the n roots, and 



m 1 m 2 ...m n 



lias the + or — sign prefixed, according as n is even or odd. 



A n _i is the sum of the coaxial minors of 1), having (n—1) 2 



constituents, each divided by the product of {n — 1) vis, and 



is equal to the sum of the products of the roots taken n—1 



together, and has the opposite sign to that of A„ prefixed, 



and so on down to A 2 , which is the sum of the coaxial minors 



of D having 2 2 constituents, each divided by the product of 



two m's, as 



/>,>- 



, and is equal to the sum of the product.- 



of all pairs of the roots, and always has the positive sign 

 prefixed. 



It thus appears that the determinant D divided by 



2 



