290 Mr; Sylvester's Elementary Researches in the Analysis of 



Arrange in cyclothemes ( — - — in number) the odd mo- 

 dulus system A, B, C, D, E. We have thus 

 ABCDE 



ACE B p. 



Let each cyclotheme be taken in the four positions given in 

 the table above, we have thus 2x4, i. e. eight arguments. 



abcde.a@y$e.abyde.ct(3c$e 

 a/3y8e, ab c d e, a /3 c 8 e, abyds 



acebd.ayefid.uyebd.acefift 

 a y e /3 8, ac e b d, ac e /3 8, aysbd 

 And each of these arguments will furnish one bipartite syn- 

 theme, by reading off, as before, the superior of each antecedent 

 with the inferior of each consequent ; and the least reflection 

 will serve to show that the same duad can never appear in two 

 distinct arguments. 



In like manner, if the modulus be 14 and seven parts be 

 taken, the bipartite synthemes, twelve in number, may be 

 expressed symbolically thus : 



1.1,1.1.1.1.1 

 +i .2.2.2.2 2.2 

 +2.1.2.1.2.1.2 



< 



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

 +A.C.E.G.B.D.F > 



l+A E.B.F.C.G.D 



+ 2.2. 1 . 2. 1 .2. 1 J 

 Nay more, from the above table, if we agree to name the ele- 



A B 



ments a * g 1 , &c, we can at once proceed to calculate each 



of the twelve synthemes in question by an easy algorithm. For 

 instance, 



(1 . 2 . 2 . 2 . 2 . 2 . 2) x (A . C . E . G . B . D . F) 

 = (Ai. Ci C 2 . E, E 2 . G x G 2 . B l B 2 .D,D 2 . F, F 2 . A 2 ). 

 And again, 



(2.1.2.1.2.1.2) x (A . E . B . F . C . G . D) 

 = A 2 . E 2 E x . B x B 2 . F 2 F x . C x C 2 . G 2 G! . D x D 2 . A! ; 



each figure occurring once unchanged as an antecedent and 

 once changed as a consequent. 



If it were thought worth while it would not be difficult, by 

 using numbers instead of letters, to obtain a general analytical 

 formula, from which all similarly constituted synthemes to any 

 modulus might be evolved. 



