200 



Prof. F. Y. Edgeworth on 



the y for which the corresponding ze-ordinate is greatest. It 

 is given therefore by the equations 



^i _ n ^ Wl — 

 ~dj > IF" 



s w 1 being what w becomes when ri is substituted for x) ; or by 

 the equations 



that is 



Whence 



dy~^ dz " U ' 



hx' + by+fz = 0, 

 gx'-\-fy + cz=0. 



— f9~~ c h > 5 J 



where H and A are minors of the discriminant 



a h a 



t> f 



f e 



IT 



Thus t— -r-as before. 

 A 



By parity of reasoning the most probable x corresponding 



TT 



to an assigned y' is ^ . Thus A = B. And by parity the 



coefficients of correlation between the other pairs xz and yz 

 are respectively proportional to G and F ; and C = B = A. 

 From the same point of view it appears that the coefficient 



of x 2 in the exponent of the expression \\ivdydz is j-. For 



suppose for a moment that there were added to w a term 

 — 2xx ; the centre of the surface being thus tranferred from 

 zero to x. The most probable x is found by solving the 

 system 



ax-\-hy-\-gz=^x y 



hx + by+fz=Q, 

 gx+fy +cz=0. 



Solving for x, we have Ax = Ax, x= -^x ; and accordingly 



fyvdydz is of the form Je~A {x ~ x)2 ' 



