of three sets of three letters each. 519 
containing = ® or 324 substitutions, So the family of 36 
groupings lead to the formation of an intransitive substitution 
== 
- Or 16 
group of i. e. 18, and of a transitive group ie 
substitutions. 
Since 9 letters may be thrown, in at x a : i. e. 280 differ- 
an ways, into nomes of 3 letters each, it further follows that by 
repeating each of the above two families 280 times we shall 
obtain new families remaining unaltered by any substitution of 
any of the nine elements inter se, and consequently indicating 
the existence of substitution-groups containing 
1.°2.3894.5.6.7.38.9 Fl bs 2h Bin Fe Guy 1:8; 9 
280 x 36 ia 280 x 4 
1. e. 86 and 324 substitutions respectively. 
~ In the above solution a little consideration will show that the 
method is essentially based on the solution of a previous question, 
viz. of grouping together the synthemes of binomial duads of two 
nomes of three letters each, which can be done in two distinct 
modes, which (if, ex. gr., we take 1.2.8, 4.5.6 as the two 
nomes in question) are represented j in the notation used above by 
Cy 
C5 4a | és Rowe TEly, So, more generally, the groupings of the 
Cg Cs 
g-nomial g-ads of 7 nomes of s oars may be made to depend 
on the groupings of the (g—l)-nomial (g—1)-ads of (r—1) 
nomes of selements each. The more general question is to dis- 
cover the groupings and their families of the synthemes composed. 
of p-nomial g-ads of r nomes of s elements, of which the simplest 
éxample next that which has-been considered and solved is to dis- 
cover the groupings of the synthemes composed of 54 binomial 
triads of 3 nomes of 3 elements each *. 
The chief difficulty of calculating @ priort the number of such 
groupings is of a similar nature to that which lies at the bottom 
of the ordinary theory of the partition of numbers, namely, the 
hability of the same groupings to make their appearance under 
distinct symbolical representations. Of this we have seen an 
éxample in the threefold representation of the principal family 
ef 4 groupings just treated of. But for the existence of this 
‘* T have ascertained, by a direct analytical method, since the above was. 
el that the number of different groupings of the ‘synthemes composed. 
of these binomial triads is 144. The number of distinct types or families 
is three, one containing 12, another 24, and the third 108 groupings. 
