452 JOURNAL OF THE WASHINGTON ACADEMY OF SCIENCES VOL. U, NO. 19 



As a second illustration let us consider the correlation between 

 products of the variables. 



Let y = XiXo and z — X\X% . 



Then from (1) we have 



Ty, = , =• (5) 



^ (c'l^ + V.' + 2r^,x.v,v,) W + v." + 2rx,'x/nW) 



The mean and standard deviation, derived from equations (2) and 

 (3), are given by 



niy = Wim2 [1 + r.Y,T, ?'i^'?] 



and (Ty = WiWo V vr + ^'2^ + ^rx^XiWd-z 



If we consider x-z' to be constant we have a special case of formula (5) 

 of considerable importance ; that is, the case of the correlation between 

 the product of two variables, y = x^Xi, and some third variable 



z = Xi'. 



For this case 



rr,r,'^'i + rx2Xi'V2 

 Tyz = ~ — • (6) 



^v i'^ + V2^ -\- 2rx,x2'Vi'V2 



Formulae (5) and (6) will be found useful in those cases in which the 

 product of two measurable linear functions is used as an index number 

 for a surface that cannot be directly measured, as is the case for 

 instance in dealing with the surface of the human body. 



The problem of finding the coefhcient of correlation between some 

 function of a set of variables and some other measured variable is 

 so common that it is advisable to consider the form taken by equation 

 (1) in this case. We would have 



y = f {xi, Xi, xn) and z = F (xi') = Xi'. 



Then 



n 



y'i Ja't'xaXi' Ca- 



a=l 



TyZ — 



(7) 



n n 



\ a=l ^=1 



