516 Prof. Sylvester on a Four-valued Function 
imperfect state. It was intended to be supplemented with a 
material addition which escaped my recollection when, after a 
long delay, the proofs of the paper passed through my hands. 
The question to which this Table refers is the following :— 
Three nomes, each containing three elements, are given; the 
number of ¢rinomial triads (1. e. ternary combinations, composed. 
by taking one element out of each nome) will be 27, and these 
27 may be grouped together into 9 synthemes (each syntheme 
consisting of 3 of the triads in question, which together include 
between them all the 9 elements). It is desirable to know :— 
lst. How many distinct groupings of this kind can be formed. 
2nd. Whether there is more than one, and, if so, how many 
distinct types of groupings. ‘The criterion of one grouping 
being cotypal or allotypal to another is its capability or incapa- 
bility of being transformed into that other by means of an inter- 
change of elements. Be it once for all stated that the question 
in hand is throughout one of combinations, and not of permuta- 
tions; the order of the elements in a triad, of a triad in a syn- 
theme, of a syntheme in a grouping is treated as immaterial. 
As we are only concerned with the elements as distributed into 
nomes, the number of interchanges of elements with which we 
are concerned is 6x 6? or 1296; the factor 6° arises from the 
permutability of the elements of each nome inter se, the remain- 
ing factor 6 from the permutability of any nome with any other. 
I find, by a method which carries its own demonstration with 
it on its face, that the number of distinct groupings is 40, of 
which 4 belong to one type or family, and 36 to a second type 
or family. 
Let the nomes be 1.2.3, 4.5.6, 7.8.9, and let 
ce, denote 1.4, 2.5, 3.6 C; denote 1.4, 2,6, 3.5 
eno! fired SRY 6y iB o4h) hep gle) Phe, 
Gog. 168, 204, B65) ey lah) de 
FG 9 7, 9, 8 
y denote 8, 9, 7 y' denote 9, 8, 7 
apts 8, 7,9 
b, denote 1.7, 2.8,3.9 6, dencte 1.7, 2.9, 3.8 
bg 5p 41 26080 eee >) A 
By ii ar en OF BAT PBB By, A ee 
“ 
1 OD 
ee OOD 
QO KC 
“ 
“ 
4, 5, 6 
B denote 5, 6, 4 B' denote 
6 5 
“ 
