668] 
243 
668. 
ON COMPOUND COMBINATIONS. 
[From the Proceedings of the Lit. Phil. Soc. Manchester, t. xvi. (1877), pp. 113, 114; 
Memoirs, ib., Ser. hi., t. vi. (1879), pp. 99, 100.] 
Prof. Clifford’s paper, “ On the Types of Compound Statement involving Four 
Classes,” [volume of Proceedings quoted, pp. 88—101 ; Mathematical Papers, pp. 1—13], 
relates mathematically to a question of compound combinations; and it is worth while 
to consider its connexion with another question of compound combinations, the application 
of which is a very different one. 
Starting with four symbols, A, B, C, D, we have sixteen combinations of the 
five types 1, A, AB, ABC, ABGD, (1+4 + 6 + 4 + 1 = 16 as before). But in Prof. 
Clifford’s question 1 means A'B'C'D', A means AB'C'D', &c.; viz. each of the symbols 
means an aggregate of four assertions; and the 16 symbols are thus all of the same 
type. Considering them in this point of view, the question is as to the number of 
types of the binary, ternary, &c., combinations of the sixteen combinations; for, 
according as these are combined, 
, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 
^o. ol types - 1 ^ ^ 19j 27> 47j 55j 78 ^ 55> 47> 27j 19j 6> ^ 1 
together. 
In the first mentioned point of view the like question arises, in regard to the 
sets belonging to the five different types separately or in combination with each other; 
for instance, taking only the six symbols of the type AB, these may be taken 1, 2, 
3, 4, or 5 together, and we have in these cases respectively 
-\ T £» 1, 2, 3, 4, 5 
No. of types = 2 2 ~2~i » 
31—2