unsymmetrical Siv-valued Function of six Letters. 373 
the basic syntheme complete the system required. The M system 
corresponding to any distribution of the imdices is the system 
which contains the synthematic arrangement of the bipartite* 
triads which can be constituted out of six things, separated in 
two sets or parts, and is unique. The L system is one of those 
which represents the synthematic arrangement of the tripartite t 
triads of nine things separated mto three sets or parts. I have 
set out above one in particular of these for the sake of greater 
clearness; but any other system having the same property will 
serve the same purpose, and a careful study will serve to show 
that the total number of L’s corresponding to a given distribu- 
tion of indices will be ( ) +. Consequently the total number 
of LM’s that we can form for a given distribution will be (_ ) 
xX1.2.3.4.5.6.7.8.9; and the number of distinct syn- 
thematic arrangements satisfying the given conditions corre- 
sponding to any assumed basic syntheme will be this number 
raised to the tenth power; and as this vastly exceeds the total 
number of permutations of fifteen things, we see, without even 
taking into consideration the diversity that may be produced by 
a change of the base, that this method must give rise to man 
distinct types of solution (arrangements being defined to belong 
to the same or different types, according as they admit or not 
of being deduced from each other by a permutation effected 
among their monadic elements), ‘The common character of all 
these allotypical aggregations, and which serves to constitute them 
into a natural order or family, consists in their being derived from 
a base formed out of five sets, such that the monopartite triads 
corresponding to the base form one syntheme, and the other 90 
synthemes each contain a conjugation of the tripartite triads 
belonging to three out of the five sets of the base with the bipar- 
tite triads belonging to the other two sets thereof. There is, 
moreover, no reason to suppose, or at all events no safe ground 
for affirming, that this family exhausts the whole possible num- 
ber of types to which the arrangements satisfying the proposed 
condition admit of bemg reduced. A further question which I 
have somewhere raised, and which brings the two problems of 
the school-girls into rapport, is the following :—* To divide the 
_ system of 91 synthemes satisfying the conditions above stated 
into thirteen minor systems, each of which satisfies the conditions 
of the old problem, 7. e. of containing all the duads that can be 
made out of the fifteen elements once and once only ;” or to put 
the question in a more exact form, to exhibit thirteen systems, 
* See note at end of paper. 
+ Some day or another a newcombinatorial caleulus'must come into being 
én furnish general solutions to the infinite variety of questions of multifa- 
riousness to which the theory of combimatorial aggregation, alias compound 
permutations, gives rise. 
