Prof. Sylvester on Tactic. \ 53 



groupings of 18 synthemes each with each other, with a view to 

 ascertain which admit of being permutable into each other, and 

 which notr 



The largest species of 108 groupings, it may be observed, is 

 subdivisible into 3 varieties, not really allotypical, of 36 each, — 

 the characteristic of those groupings which belong to the same 

 variety being that they permute exclusively into each other 

 when the permutations of the elements are confined to perturba- 

 tions of the order of the elements in the same nome or nomes, 

 and the different nomes are subject to no interchange of elements 

 between themselves. 



Just so the species of 36 groupings of trinomial triads, treated 

 of in my preceding paper, subdivides into 3 varieties or sub- 

 families characterized by a similar property. 



The total number of modes of subdivision of 9 elements 

 between 3 nomes being 280, it follows, from considerations of 

 the same kind as stated in the May Number of the Magazine, 

 that there exist transitive substitution- groups belonging to 9 ele- 

 ments of 



tt(9) tt(9) tt(9) 



280x12' 280x24' 280x108' 

 that is, 108, 24 and 12 substitutions respectively. 



Again, let us consider the question of forming the synthemes 

 of the triads of a single nome of 9 elements into groupings where 

 every triad shall be found without repetition. We may obtain 

 such groupings by choosing arbitrarily any one of the 280 sets 

 of 3 nomes into which the 9 elements may be segregated*, and 

 then forming one syntheme with the three monomial triads 

 (corresponding to such set so chosen), 18 synthemes (in any one 

 of the 144 possible ways) of exclusively binomial triads, and 9 

 synthemes (in any one of the 40 possible ways) of exclusively 

 trinomial triads ; we shall thus obtain in all 280 x 144 x 40, or 

 1,612,800 solutions of the question proposed ; I mean 1,612,800 

 groupings, all satisfying the imposed condition, and reducible to 

 6 general, comprising respectively 



4x12x280 4x24x280 4x108x280 36x12x280 

 36 x 24 x 280 36 x 108 x 280, 



* 280 is also evidently the number of synthemes of triads belonging to 

 one nome of 9 elements. In general the number of r-ads belonging to 

 one nome of nm elements is 



ff(mn-l)ff((m-l)n-l)7r((m-2)n-l)...»r(n-l) 



(*■(» - 1 )) m T((m _ 1 ) n y\{m - 2)») . .. *■(») 



t The above genera must not be confounded with types or species. (In 

 my preceding communications I may inadvertently have used the word 



