Combinatorial Aggregation. 293 



The syntheme in question is therefore 



A 2 . B 4 A 4 . B 2 B 3 . C 2 B x . C 4 C x . A 3 C 3 . A v 

 and so on for all the rest, the rule being that 

 r: s — r .s + 1 +r + 2.j+ 3. 

 Now, as before, it is evident that if we look only to conti- 

 guous terms, the above table of position may be extended to 

 any number of odd terms, simply by repetition of the second 

 and third figures in each syzygy ; and hence the rule for ob- 

 taining the bipartite synthemes to the modulus 4 xp is appa- 



7— 1 

 rent. For instance, let p = 7, there will be 8 x — - — , i. e. 



29 



8x3 of them denoted as follows : — 



f A.B.C.D.E.F.g] f 1-1. LI- LI- 1+2. 1.2.1.2.1.2" 

 J . I I +1.2.3.2.3.2.3+2.2.4.2.4.2.4 



+ A.C.E.G.B.D.F r x j +1.3.2.3.2.3.2 + 2.3.1.3.1.3.1 

 + A.E.B.F.C.G.DJ L+ 1 - 4, . 4 . 4, - 4, «4. 4 * + 2.4.3.4.3.4.3_ 



As an example of the mode of development, let us take the 

 term 



A.E.B.F.C.G.Dx2.4.3.4.3.4.3 



2 . 4 . 3 . 4 . 3 . 4 . 3 = (2 : 4, 4 : 3, 3 : 4, 4 : 3, 3 : 4, 4 : 3, 3 : 2) 

 _/ 2.11 4.41 3.1 1 4.41 3.H 4.4l 3.3\\ 

 ~V+*.3J +2.2J +1.3J + 2.2 J +1.3/ +2.2 J + 1.1 J / 



A.E.B.F.C.G.D=A.E,E.B,B.F,F.C,C.G,G.D,D.A, 



and the product 



_/A 2 .E x E 4 .B 4 Bs.Fj F 4 .C 4 Cg.^ G 4 .D 4 D 3 .A 3 \ 



-VA 4 .E 3 E 2 .B 2 B..F3 F 2 .C 2 fcJ.G, G 2 .D 2 D^aJ'- 



Let the modulus be 6 x 3, as before, give ajixed cyclic order 

 to the constituents of each part, and each will admit of being 

 exhibited in six positions. 



Write similarly as before, 



a l b x Cj 

 a <2 o<2 c 2 



a 3 "3 C 3 



« 4 b 4 c 4 

 a s h C 5 



a 6 "6 C 6> 



and take the odd places of each antecedent with the even 

 places of each consequent; it will now be seen that 

 r\ s — r + 2:s + 2 = r + 4:s + 4, 



and the number of independent permutations is — '—- = 2.6; 



