100 PROF. CAYLEY ON COMPOUND COMBINATIONS. 



combinations of the five types i. A, A B, ABC, ABCD 

 (1+4 + 6 + 4+1 = 16 as before) . But in Prof. Clifford's 

 question i 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. Consi- 

 dering them in this point of view, the question is as to 

 the number of types of the binary, ternary, &c. combi- 

 nations of the sixteen combinations ; for, according as 

 these are combined. 



No. of types = ^>^^3. 4. 5. 6, 7, 8, 9,10,11,12,13,14,15 

 1,4,6,19,27,47,55,78,55,47,27,19,6,4,1 

 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 i, 2, 3, 4, or 5 together, and we have in these 

 cases respectively 



No. of types 



_i. 2,3,4, 5 



I, 2, 2, 2, I 



as is very easily verified ; but if tlic number of letters A, B . . . 

 be greater (say this = 8), or, instead of letters, writing the 

 numbers 1, 2, 3, 4, 5, 6, 7, 8, then the question is that of 

 the number of types of combination of the 28 duads 12, 13, 

 ...78, taken i, 2, 3, ...27 together, a question presenting 

 itself in geometry in regard to the bitangents of a quartic 

 curve (see 'Salmon's Higher Plane Curves,' Ed. 2 '(1873), 

 pp. 222 et seq.) : the numbers, so far as they have been 

 obtained, are 



No. of types^^'"'3> 4 24,25.26,27 



I. 2, 5, II II. 5. 2, I 



It might be interesting to complete the series, and, more 

 generally, to determine the number of the types of combi- 

 nation of the \n{)i—i) duads of « letters. 



