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

 and so in general, if there be 2 m constituents in a part, the 

 number of independent permutations is - = 4ra. 



2~ 



The rule for the formation of the table will be apparent on 

 inspection. I suppose only three parts, as the rule may 

 alwa}'s be extended to any number by reiteration of the 

 second and third terms. The table will be found to resolve 

 itself naturally into four parts, each containing m lines. 



Let m = 1, we have 



{ 



So that x, going through all its values from 1 to m, the gene- 

 ral expression for the four parts is 



<st(\.x.2x— \ + \.m+x.2x \ 

 \+ 2.x.2x + 2.W + x.2x — 1/ 

 To show the use of this formula, let us suppose that we have 

 seven parts, each containing ten terms, the general expression 

 for the bipartite duad synthemes is 



( \.x.2x — \.x.2x — l.x.2x — 1 

 A.B.C.D.E.F.G^I J + 2.x.2x.x.2x.x.2x 

 + A.C.E.G.B.D.F > x'Zi L JJ^^x.5 + ^.2x.5^.2T 



+ A.E.B.F.C.G.D J r .-— -z T j— « r t^~ 5— 



^ + 2.5 + x.2x— 1.5 + x.2x — l.5 + x.2x— 1 



