292 Mr. Sylvester's Elementary Researches in the Analysis of 



each antecedent with the second and fourth respectively of 

 each consequent ; we have accordingly, 



a 3 .b 4 b R .c 4 c 3 .a 4 . 

 It is apparent that the same combinations will recur if any 

 two contiguous parts revolve simultaneously through two steps; 

 or in other words, that A r . B s = A r+2 . B s+2 , where ft is any 

 number, odd or even. 



Symbolically speaking, therefore, as regards our table of 

 position, r: s ■= r + 2 : s + 2, or more generally, 



= r + 2 ± 4 i : s + 2 + 4 i. 

 So that 1:1=3:3 2:1=4:3 



1:2 = 3:4 2.2 = 4:4 



1:3 = 3.1 2.3 = 4:1 



1:4 = 3.2 2.4 = 4:2. 

 There are therefore no more than eight independent un- 

 equivalent permutations to every pair of parts. Now inspect 

 the following table of position : — 



i . 1 . i 2.1.2 



1.2.3 2.2.4 

 1.3.2 2.3.1 



1.4.4 2.4.3 



It will be seen that in the first and second, second and 

 third, third and first places, all the eight independent per- 

 mutations occur under different names', the law of forma- 

 tion of such and similar tables will be explained in due time; 

 enough for our present object to see how, by means of this 

 table, we are able to obtain the bipartite synthemes to the 

 given modulus 4x3; the number according to our formula 



3 — 1 

 is 2 x 4 x — - - — = 8, and they may be denoted symbolically 



as follows: — 



(A R pv /l. 1.1 + 1.2.3 + 1.3.2+ 1.4. 4\ 

 vA-o.^J y + 2.1.2 + 2.2.4 + 2.3.1 +2.4.3/' 



Each of the eight terms connected by the sign of + gives a di- 

 stinct syntheme ; ex. gr. let us operate on 



A.B.C x (2.3.1). 

 2.3.1 denotes 2.3 3.1 1.2. 



2 . 3 gives rise to 2. 3+ 1 +2+2.3+3 

 = 2.4 + 4.2. 



3 . 1 gives rise to 3. 1+1+3+2. 1 + 3 

 = 3.2 + 1.4. 



1 . 2 gives rise to 1 . 2+1 + 1 + 2 . 2+3 

 = 1.3+3.1. 



