196 



Whence 



Prof. F. Y. Edgeworth on 



Pl =A(l-p 23 2 ) 



9i2 = ^(p2sPn—pi2) ; 



with corresponding values for the other coefficients. 



It remains only to determine A. This is effected by the 

 equation A = P 1# Employing the values of p lf p 2 &c, q i2 , &c, 

 which have just been found, 



P 1= A(l-^ 13 2 ) A(pi2p23~pn) 

 &(pi2P2Z-Pn) A(l-/> 12 2 ) 

 whence 



A=A 3 {(l-p 13 s )(l -prft-ipnPK-pizYl ; 



and A is determined in terms of thep's, of which the numerical 

 values are supposed to have been ascertained. The whole 

 system of coefficients is therefore determined numerically. 



Example. — Let x±, x 2 , x s respectively represent deviations 

 of stature, cubit, and height of knee. The coefficients of cor- 

 relation for each pair are p 13 = *8, p 13 =-9, p 2 3 == '^ > as ascer- 

 tained by Mr. Galton (Proc. Roy. Soc. 1888, Co-relations, 

 Table V.). To find the coefficients of the expression 



Pi^l* +p 2 Xc? +p&si* + 2q 12 x { x 2 + 2q u x 1 x 3 + 2q 2B x 2 x, 



which is the exponent of the expression for the probability 



that 



any par 



rticular values of 



a?!, a 



> tl '2> 



should concur; or, in 



other words, the equation of the ellipsoid of equal probability 

 (the final constant being omitted). 



Here for the reciprocal of the discriminant A we have 



A'= 



A-8, 

 A-9, 



Whence 



jv-=A 



A-8, 

 A, 



A-8, 



•8 



A-9 



A-8 

 A 



'8, 



A(l--64)=A-36, 



?12 = A 



= A(-72--8) = -A-08. 



•8, -8 

 1, -9 

 By parity p 3 = A'19, p s = A'36, q ld =-A'26, q 2Z =-A-Q8. 



A-36, -A-08, -A-26 



Thus A = -A-08, A- 19, -A-08 



-A-26, -A-08, A-36 



