260 MR. G. UDNY YULE ON THE ASSOCIATION 



(A) = (a) 



(B) = (/8), &c., 

 (AB) = (a/?) 



(aB) = (A/3), &c., 



(ABC) = (aygy) 

 (ABy) = (a/SC), &C., 



and so on with groups of any order. 



I shall return later to this case and to the properties that this equality of con- 

 traries produces. 



5. It may be noted at this point that groups may often be rapidly expanded by 

 using ABC, &c., as " elective operators" in BOOLE'S sense, and using the general law 

 of multiplication of operators, with the special conditions 



UA = A, 



i.e., selecting out the universe of discourse, and then selecting out the A's from it, is 

 the same thing as selecting out the A's at once, and the " index law " 



A = A, 



i.e., repeating the operation of selecting out the A's has no effect on the objects 

 included. 



To denote that the letters are being used as operators we will use square brackets. 

 Thus 







[ABy] = [U - a] [U - ft [U - C] 



= [U 3 ] - [IPa] - [U 2 j8] - [U'C] + [Ua/3] + [UaC] + [UC] - [aftC] 

 or 



(ABy) = (U) - (a) - 08) - (C) + (aft) + (aC) + (/8C) - (aftC). 



We only mention the process as it affords such a rapid and easy means of 

 expansion. The results obtained by its use can always be obtained at a little greater 

 length by an elementary process of step-by-step substitution. 



6. Before proceeding to the consideration of association and so forth, it seems 

 necessary to discuss somewhat fully the general relations subsisting between the 

 frequencies of different groups, and the number of independent frequencies of any 

 order. Suppose, for example, we are dealing with three attributes (A, B, C and 

 their contraries). Twelve second order and eight third order groups can be formed 

 from these. It might appear then that if the frequencies of the second order groups 

 were given, there would be a sufficient number of equations to determine the 

 frequencies of the third order groups. As a matter of fact this is not so ; the twelve 

 second order frequencies do not form independent data, and the question arises, How 

 many are independent ? or, in general, how many independent frequencies or groups 

 are there in the mth order groups produced from n variables ? i.e., how many of these 



