Combinatorial Aggregation, 289 



Now we may form a cyclic table of positions as below : 



ABC 

 1 1 1 



1 2 2 



2 1 2 

 2 2 1 



Here the numbers in each horizontal line denote the synchro- 

 nic positions of the parts. 



On inspection it will be discovered that A will be found in 

 each of its two positions, with B in each of its two; similarly 

 B with C, and C with A. In fact the four permutations, 

 1.1 1.2 2.1 2.2, occur, though in different orders, in any 

 two assigned vertical columns. 



Now develope the preceding table, and we have 



ace a df b cf b de 



b df bee ade a cf; 

 and these being read off (the superior of each antecedent with 

 the inferior of each consequent*) must manifestly give the 

 four independent bipartite synthemes which we were in quest 

 of, videlicet 



{a.d cfe.b), (a.c d.ef.b), {b.dc.ef.d), (b.c d.fe.a); 

 these four, together with the synthemefirstdescribed(d!.6c.rf<?./), 

 constitute a duad synthematic total to modulus 6. 



Before proceeding further let us take occasion to remark 

 that the foregoing table of positions may evidently be extended 

 to any odd number of terms by repetition of the second and 

 third places, as seen in the annexed tables of position. 



i .1 . 1. 1 . i i . i . i . i . i . i . i 



1 .2.2.2.2 \ .2 .2.2.2 .2.2f 

 2.1.2.1.2 2.1.2.1.2.1.2 



2.2.1.2.1 2.2.1.2.1.2.1 



Now let 10 be the modulus. 



As before divide the elements into five parts, which call 

 A, B, C, D, E. 



The unipartite duads fall into a single syntheme ; the eight 

 remaining bipartite synthemes may be found as follows : — 



* Any other fixed order of successive conjunction would answer equally 

 well. 



t It will not fail to be borne in mind that in operating with these tables 

 only contiguous elements are taken in conjunction : the first with the second, 

 the second with the third, the third with the fourth, &c, and the last with 

 the first; no two terms but such as lie together are in any manner conju- 

 gated with one another. 



Phil. Mag. S. 3. Vol. 24. No. 159. April 1844. U 



