G. U. YuLE 131 



and also 



(ßCD ... MN) = ^— m{('){D) ... (il/)(iV)l, 



\ve niu«t havc 



(aBCD . . . MN) = {BCD . . . MN) - {ABCD . . . MN) 



= ^. m{C){D) ... {M){N)\ [N-{A)] 



= ^M{B){C){D)...{M){N), 



and so od. The nuniber of clas.s-frequencie.s to be te.stcd in ordcr to demonstrate 

 the existence of cunipletc independeuce is, of cour.se, the same as bofore, viz. 



It should be noted as a consequence of these results that the definition of 

 "complete independence " given on p. 127 i.s redundant in its terms. It is quite 

 true that if complete independence subsist for a series of attributes every possible 

 pair must exhibit independeuce in every possible sub-universe as well as in the 

 iiniverse at large, but it is not necessary to apply the criterion of independeuce to 

 all these possible cases. In the case of three attributes for instance the criterion 

 of independence need only be applied to four frequencies, as \ve have just seen, in 

 Order to demonstrate complete independeuce ; it cannot then be necessary, as 

 suggested by the definition, to test niue different associations, viz. 



I ^5 I \ AB\G\ 



\AC\ \AC\B\ 



\ BC \ I 56' I 4 I 



in the uotation of my memoir on Association (au expression like | .45 | G \ 

 specifying " the association between A and B in the universe of (7's"). It is 

 in fact only necessary to test |-45|, I^CI, \BG\, and AB\G\ (or one of the 

 other three partial associations in positive universes). If these are zero, the 

 remaining associations must be zero also ; for we are given 



(ABG)^-^^ {AG){BG) = ^.^ (A) (B) (G), 

 ■.(ABG) = ^^(AB)iBG) 



= ^i^(^5)(46') = etc. 



i.e. I AG\ B\,\ BG \ A |, etc. are zero. Quite generally, it is ouly necessary, if the 

 testing be suppcsed to proceed fmm the second order classes upwards, to test one 

 of all the possible partial associations corresponding to each positive class. If 

 there be four attributes ABGD, the six total associations | AB |, | AG\, \ AD \,\ BG \ 



17-2 



