370 Prof. Cay ley on a Problem of 



Only when the insoluble salts (as, for instance, carbonate of 

 manganese by washing with hot water) alter their constitution 

 and become basic, is the magnetism of their atom smaller, as in 

 the case of the dissolved basic ferric salts. 



As regards theoretical conclusions from the present results of 

 observation, a paper to be forthwith published in PoggendorfFs 

 Annalen must be consulted. 



LIII. On a Problem of Geometrical Permutation. 

 By Professor Cayley, F.R.S * 



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 inflexions, 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 any 

 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 



„ „ 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 ; 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 on y re- 

 maining modes are therefore those which have 3 figures, 1 T gure, 

 or figure in loco. 



* Communicated by the Author. 



