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ON A PROBLEM OF GEOMETRICAL PERMUTATION.
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for any two inflexions are in a line with a third inflexion; and if the figures which
belong to the first two inflexions are in loco, then the figure belonging to the third
inflexion will be in loco; that is, there will be 3 figures in loco. The only remaining
modes are therefore those which have 3 figures, 1 figure, or 0 figure in loco.
First, if three figures are in loco, these, as just seen, will be the figures which
belong to three inflexions in a line. Suppose the figures are 1, 2, 3 ; then the inflexion
originally denoted, say by the figure 4, may be denoted by any one of the remaining
figures 5, 6, 7, 8, 9; but when the figure is once fixed upon, then the remaining
inflexions can be denoted only in one manner. Hence when the figures 1, 2, 3 remain
in loco there are 5 modes; and consequently the number of modes wherein 3 figures
remain in loco is 5 x 12, = 60.
Next, if only a single figure, suppose 1, remains in loco, the triads which belong
to the figure 1 are 123, 147, 159, 168; and there is 1 mode in which we simultaneously
interchange all the pairs (2, 3), (4, 7), (5, 9), (6, 8). (Observe that the triads
123, 147, 159, 168 here denote the same lines respectively as in the primitive denotation,
the figure 1 remains in loco, but the figures belonging to the other two inflexions on
each of the four lines are interchanged.) There are, besides this, 2 modes in which
the figures (2, 3), but not any other two figures, are interchanged ; similarly 2 modes
in which the figures (4, 7), 2 modes in which the figures (5, 9), 2 modes in which
the figures (6, 8), but in each case no other two figures, are interchanged; this gives
in all 1+2 + 2 + 2 + 2, =9 modes. There are besides, the figure 1 still remaining
in loco, 18 modes where there are no two figures (2, 3), (4, 7), (5, 9), or (6, 8) which
are interchanged: viz. the figure 2 may be made to denote any one of the inflexions
originally denoted by 4, 5, 6, 7, 8, or 9. Suppose the inflexion originally denoted by 4;
3 will then denote the inflexion originally denoted by 7: it will be found that of
three of the remaining six inflexions, any one may be denoted by the figure 4, and
that the scheme of denotation can then in each case be completed in one way only.
This gives 6x3, =18, as above, for the number of the modes in question; and we
have then 9 + 18, =27, for the number of the modes in which the figure 1 remains
in loco; and 9 x 27, = 243, for the number of modes in which some one figure remains
in loco.
Finally, if no figure remains in loco, the figure 1 will then denote some one of the
inflexions originally denoted by 2, 3, 4, 5, 6, 7, 8, 9. Suppose it to denote that originally
denoted by 2; 2 cannot then denote the inflexion originally denoted by 1, for if it
did, 3 would remain in loco: 2 must therefore denote the inflexion originally denoted
by 3, or else some one of the inflexions originally denoted by 4, 5, 6, 7, 8, 9. It
appears, on examination, that in the first case there are 4 ways of completing the scheme,
and in each of the latter cases 2 ways; there are therefore in all 1 x 4 + 6 x 2, = 16
ways; that is, 16 modes in which (no figure remaining in loco) the figure 1 is used
to denote the inflexion originally denoted by 2; and therefore 8x16, =128 modes, for
which no figure remains in loco. This completes the investigation of the numbers
1, 60, 243, and 128, which together make up the total number 432 of the modes of
denotation of the nine inflexions.
