48 Prof. Sylvester on Tactic. 



it will be completely and distinctively denoted in brief by the 



twelve forms arising from the development of 

 a b c a c b 



b c a and b a c ; videlicet 

 cab c b a 



(1) 



(2) 



(3) 



(4) 



(5) 



(6) 



1 2 3 



2 3 1 



3 1 2 



2 1 3 



13 2 



3 2 1 



2 3 1 



3 1 2 



12 3 



13 2 



3 2 1 



2 1 3 



3 1 2 



1 2 3 



2 3 1 



3 2 1 



2 13 



1 3 2 



(7) 



(8) 



(9) 



(10) 



(11) 



(12) 



13 2 



2 1 3 



3 2 1 



2 3 1 



12 3 



3 1 2 



2 13 



3 2 1 



13 2 



12 3 



3 1 2 



2 3 1 



3 2 1 



13 2 



2 13 



3 1 2 



2 3 1 



1 2 3 



which we may for facility of future reference denote by 



7T, 7T 2 7T 3 7T 4 7T 5 . . . 7T, 2 . 



Now as regards the t)^pes : since the order of the elements in 

 one nome is entirely independent of the order of the elements in 

 any other, it is obvious that it is not the particular form of P or 

 of P which can have any influence on the form of the type, but 

 solely the relation of P and P to one another. In order then to 

 fix the ideas, I shall for the moment consider P equal to 

 12 3 



2 3 1 



3 1 2 



This at once enables us to fix a limit to the number of distinct 

 types. In the first place, the essentially distinct forms of the 

 first column in P, with respect to that of P, may be sufficiently 

 represented by taking the two columns identical, or differing by 

 a single interchange, or else having no two elements in the same 



place. Hence P, so far as the ascertainment of types is con- 

 cerned, may be limited to the six forms following : — 



(*) (y) W 



123 213 231 



231 132 312 



312 321 123 



(0) 



(S) 



in) 



13 2 



2 3 1 



2 1 3 



2 13 



12 3 



3 2 1 



3 2 1 



3 1 2 



1 3 2 



But again, since (j3) and (rj) are each derivable from («) (the 

 assumed form of P) by an interchange of two columns inter se, 



