Geometrical Permutation. 371 



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 remain- 

 ing 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 origi- 

 nally 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 in- 

 flexions, 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 

 9x27, =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 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 origi- 

 nally 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 

 1x4 + 6x2, =16 ways ; that is, 16 modes in which (no figure 

 remaining in loco) the figure 1 is used to denote the inflexion 



2B2 



