376 On the unsymmetrical Siz-valued Function of six Letters. 
The discussion of the properties of this Table, and the classi- 
fication of the eight aggregates into natural families, must be 
reserved for a future occasion. 
Note.—A triad is called tripartite if its three elements are 
culled out of three different parts or sets between which the 
total number of elements is supposed to be divided; bipartite 
if the elements are taken out of two distinct sets ; unipartite if 
they all lie in the same set. The more ordinary method for 
the reduction of synthematic arrangements from a given base 
to a linear one which I employ, consists in the separate synthe- 
matization inter se of all the combinations of the same kind as 
regards the number of parts from which they are respectively 
9 
drawn. ‘Thus, ex. gr., if the distribution of the oo dae 2 — 
triads to the base 30 into 20609 synthemes be required, this 
2 
may be effected by dividing the 30 elements in an arbitrary 
manner into 15 parts, each part containing 2 elements. These 
15 parts being now themselves treated as elements, are first to be 
conjugated as in the old 15-school-girl problem, and each of these 
7 -conjugations can be made to furnish 6 synthemes containing 
exclusively bipartite triads. The same 15 parts are then to he 
conjugated as in the new school-girl problem, and the 91 conju- 
gations thus obtained will each furnish 4 synthemes, containing 
exclusively the tripartite triads. These bipartite and tripartite 
synthemes will exhaust the entire number of triads of both 
kinds, and accordingly we shall find 
7x6491 x 4=406 
_ 29 x 28 
OnE 
A syntheme, I need scarcely add, is an aggregate of combina- 
tions containing between them all the monadic elements of a 
given system, each appearing once only. In the more general 
theory of aggregation, such an aggregate would be distinguished 
by the name of a monosyntheme. A disyntheme would then 
signify an aggregate of combinations containing between them 
the duadic elemienits; each appearing once only, and so forth. 
Thus the old 15-school-girl question in my nomenclature would 
be enunciated under the form of a problem “to construct a triadic 
disyntheme, separable into monosynthemes to the base 15 ; ” the 
new school question, as a problem “ to divide the whole of the 
triads to base 15 into monosynthemes ;”’ the question which 
connects the two, as a problem “to exhibit the whole of the triads 
to base 15 under the form of 13 disynthemes, each separated 
into 7 monosynthemes. 
