494 
ON A PROBLEM OF GEOMETRICAL PERMUTATION. 
[368 
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.