372 Prof, Sylvester on the Historical Origin of the 
fl, b, Co 
three sets a, 6, cs; and in respect of 1, 4, conjugate the two 
a, b. Ce 
gh a, b, ¢, 
remaining sets 
4 04 Ca 
From the ternary conjugation form the nine arrangements, 
Ug Ag as by by Os Co Cy Cs 
a2 as bs by bs Cs Cy ¢, ds 
fy Ag Ce by bs as Co Cy Og 
a3 bs ds by c3 55 Cg 43 ¢s 
dy bs bs by os ¢5 C9 a ds 
Aq bs. Cs by C3 ds Co a; b; 
lg Cg As be as Ob; Co by Cz 
a, Cy 6. bg fg C. Cy bs as 
fly Cy Cs by a3 as Co by 6, 
which call 
L, L, L, i, L; L, L, L, Lig. 
Again, from the binary conjugation, form the nine arrange- 
ments, 
a, b, e4 ay b,c, 
a, b, b, Gy C, Cy 
a 6, ay C) by Cy 
QC) Ca a, 6, 5, 
ay C, b, ay by ey 
@ Cy ay b, bs 4 
b, cy C, a, b, ¢, 
b, c, by Ay a C4 
b, cy a, by ey 
which call 
M, M, Mg M, M; Mg M, Mg Mog. 
Now combine the L with the M system, each L with some 
M im any order whatever; the 9 combinations or appositions 
thus obtaimed will give a batch of 9 synthemes; and proceeding 
in like manner with each of the 10 distributions of the indices 
1, 2, 3, 4, 5, we shall obtain 90 synthemes, which together with 
