[367 
368] 
493 
f ¥: for 
ue of cf) 
! 6 2 > a 2 . 
;ermined 368. 
I to the 
curve. 
ON A PROBLEM OF GEOMETRICAL PERMUTATION. 
[From the Philosophical Magazine, vol. xxx. (1865), pp. 370—372.] 
It is required to find in how many modes the nine points of inflexion of a cubic 
curve can be denoted by the figures 1, 2, 3, 4, 5, 6, 7, 8, 9, in such wise that the 
twelve lines, each containing three points of inflexion, shall be in every case denoted 
by the same triads of figures, say by the triads 
123, 
147, 
159, 
168, 
456, 
258, 
267, 
249, 
789, 
369, 
348, 
357. 
We may imagine the inflexions so denoted in one particular way, which may be 
called the primitive denotation; then in auy other mode of denotation, a figure, for 
example 1, is either affixed to the inflexion to which it originally belonged, and it 
is then said to be in loco, or it is affixed to some other point of inflexion. This 
being so, the total number of modes is = 432; viz. this number is made up as 
follows: 
9 figures in loco 1 
3 „ „ 60 
1 figure „ 243 
0 „ „ 128 
432 
There is of course only one mode wherein the nine figures remain in loco. It 
may be seen without much difficulty that there is not any mode in which 8, 7, 6, 5, 
or 4 figures remain in loco. There is no mode in which only 2 figures remain in loco;