OF ATTRIBrns |\ STATISTICS. 



273 



That \B to say, "The excesses of (AB) and (aft) alxive (AB) and (aft) tl , and of 

 (A) and (B) above (A.ft) and (B), are all equal, and equal to the ratio of the 

 difference of the cross-products to the number of observations." This theorem seems 

 to me rather remarkable. I can find no similar relation for the sum of the cross- 

 products so as to give a complete physical meaning to Q. 



23. Next let us determine any one of the second order frequencies, e.g. (AB), in 

 t.-i ins of Q and the first order frequencies. 

 If we write 



1 + Q 



we have 

 Now 



(2) 



(3;. 



whence 



(aB) = (B) - (AB) 

 (A) = (A) - (AB) 

 (aft) = (ft) - (A) + (AB). 



(AB)'(l - K ) - (AB) { * (U) + (l - K) [(A) + (B)] } + (A)(B) = ( 



which is a quadratic for (AB). 

 Now let 



(A) -_(.) 



(B) - 09) 

 (B) + (0) ~ 



(4). 



where s, *, may be called the surpluses of A and B. It follows that 



- -- (5) 



and similarly for (B) and (y8). In terms of these symbols the quadratic may be 



written 



(AB)' - (AB)(U) ~ i-* 1 - - 





whence 



CXCIV. ^ 



(*, s a j* 

 2 N 



