Combinatorial Aggregation. 287 



For instance, if the elements be called after the letters of 

 , ,,, , (a.bb.cc.dd.ee.a\ . ,. . 



the alphabet, we have \ a . c c , e e ,b b .d d.a)> tI,e bisyntlie- 



matic total to modulus 5; and in like manner 

 a.b b .c c.d d.e e .ff. g g • a "J 



a.c c ,e e .g g .b b. d d .ff. a >the total to modulus 7. 

 a.d d.g g.c c .ff. bb.ee.aj 



In general, if n be the modulus, the number of duads is 



n. — - — ; n being even, — duads go to each syntheme, and 



therefore the total contains (n — 1) of these. If (is) be odd, 

 then, since always n duads go to a bisynlheme, the number of 



such in the total is — - — . 

 2 



Before proceeding to the solution of the problem first pro- 

 posed, let us investigate the theory of diplothematic arrange- 

 ment. Here we shall find another term convenient to employ. 

 By a Cyclotheme, I designate a fixed arrangement of the ele- 

 ments in one or more circles, in which, although for typo- 

 graphical purposes they are written out in a straight line, the 

 last term is to be viewed as contiguous and antecedent to the 

 first; the recurrence may be denoted by laying a dot upon the 



two opened ends of the circle; a.b ,c .d . e will thus denote 



a cyclotheme to modulus 5 ; a.b . c . d .e ,f. g.h .k the same 



to modulus 9 ; so also is a . b . c, d . e .f g .h.Jc a cyclotheme 

 of another species to the same modulus. In general the num- 

 ber of terms will be alike in each division of a cyclotheme. 



Now it is evident that every cyclotheme, on taking together 

 the elements that lie in conjunction, may be developed into a 

 diplotheme. Thus 



1.2.3=1.2 2.3 3.1 



i . 2 . 3. 4=1. 2 2.3 3.4 4.1, 



(1.2 2.3 3. 1\ 

 4.5 5.6 6.4 I. 

 7.8 8.9 9.7/ 



9 Hence we shall derive a rule for throwing the duads of any 

 system into bisynthemes. 



Let m = 3, we have simply abc 



m = 5, we write a . b . c . d . e 



a . c .e .b . d 



