52 Prof. Sylvester on Tactic. 



vised to satisfy himself by actual trial of the fact) that the effect 

 of interchanging N, and N 3 with each other is to convert 



u ft y 8 e 

 into 



a 8 7 ft e. 



From these two calculations it follows that the effect of any 

 permutation between N„ N 2 , N 3 is to produce a permutation in 

 ft, 7, 8 inter se, but will leave a and e unaltered*. Hence then 

 we have arrived at the goal of our inquiry, having demonstrated 

 that 



j V« 



indicates one type, 









1 Va 1 



Va 





V« 



1 V/3 |, 



v 7 



> 



vs 



each of them another the same type, and 



1 Va 1 



' 



1 Ve 







a third type, — and bearing in mind that 



(a) belongs to 7r, exclusively, 

 ( e ) » T& *a m 



ifi) » 7r 7> '"'Si ^9 J> 



(7) » *4> *5i ^6 » 



(S) « 77"io> w'nj w w „ 



and that each form of 7r comprehends 12 groupings due to the 

 12 forms of Yen, we are enabled to affirm that the total number 

 of groupings of the binomial triads of 3 nomes of 3 elements 

 each is 144, and that the number of types or species between 

 which these 144 are distributed is 3, comprising 12, 24, and 

 108 respectively, — a conclusion which it would almost have 

 exceeded the practical limits of human labour and perspi- 

 cuity to have established by the direct comparison of the 144 



* This result, by the aid of a fine observation, may be more rapidly 

 established uno ictu (I mean by one calculation instead of two) as follows 

 Let Nj N 2 N 3 be made to undergo a cyclical interchange, then it will be 

 found that ft y, 8 also undergo a cyclical interchange, whilst a and e 

 remain unchanged. This proves that /3, y, 8 are only different phases of 

 the same type, which is sufficient ; for as regards a and e, the fact of the 

 number of individuals which they represent being unequal inter se, and 

 also unequal to the number contained in |3, y, 6, renders it « priori impos- 

 sible to allow that they can either pass into each other or into the forms 

 P> y> S, by virtue of any interchange among the elements. 



