unsymmetrical Six-valued Function of sia Letters. 871 
when any interchange is effected between the letters which enter 
into them; so that any oue of these is a function of six letters 
having only six values. I conceive that, after this reference, no 
writer on the subject wishing to specify the function in question 
would hesitate to call it after my name. 
I may also take occasion to observe that, in connexion with 
my researches in combinatorial aggregation, long before the 
publication of my unfinished paper in the Magazine, I had 
fallen upon the question of forming a heptadic aggregate of 
triadic synthemes comprising all the duads to the base 15, 
which has since become so well known, and fluttered so many a 
gentle bosom, under the title of the fifteen school-girls’ problem ; 
and it is not improbable that the question, under its existing 
form, may have originated through channels which can no longer 
be traced in the oral communications made by myself to my 
fellow-undergraduates at the University of Cambridge long years 
before its first appearance, which I believe was in the ‘ Lady’s 
Diary’ for some year which my memory is unable to furnish. 
In order to relieve this notice from the mere personal cha- 
racter which it may thus far appear to bear, I will state another 
question concerning the combinatorial aggregation of fifteen 
things which may serve as a pendant to the famous school-girl 
problem. 
The number of triads to the base 15 is ae =5~x9l1. 
Let it be required to arrange these into 91 synthemes, in other 
words, to set out the walks of 15 girls for 91 days (say a quarter 
of the year) in such a manner that the same three shall never all 
come together more than once in the quarter. Of the various 
ways in which it is probable this problem may be solved, the 
following deserves notice. Let 15 letters be arbitrarily divided 
into 5 sets, viz. 
Mb, 6,3 Ag bg lg3 Wg bgeg3 My by Cy; 5 5 C5 
The sets as they stand will represent one of the 91 arrangements 
sought for, which I call the basic syntheme. The remaining 90 
may be obtained as follows in 10 batches of 9 each. Write down 
the 10 index distributions following :— 
123;45 145;28 
hee 3 6 2°38 4; 15 5 
126;3 4 2 3°53; 1 4 
134; 25 2465;18 
1835; 2 4 § 45: 1.2 
Take any one of these distributions, as for instance 2 35; 1 4, 
and proceed as follows:—In respect of 2, 3, 5, conjugate the 
2B2 
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