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XX. Remark on the Tactic of 9 Elements. By J. J. Sylvester, 

 M.A., F.R.S., Professor of Mathematics at ths Royal Military 

 Academy, Woolwich*. 



AT the end of my preceding paper in this Magazine for July, 

 I hazarded an opinion that any grouping of 28 synthemes 

 comprising the 84 triads belonging to a system of 9 elements, 

 might be regarded as made up of 1 syntheme of monomial triads, 

 18 synthemes of binomial triads, and 9 of trinomial triads, the 

 denominations (monomial, binomial, and trinomial) having refer- 

 ence to a duly chosen distribution of the 9 elements into 3 nomes 

 of 3 elements each. This conjecture is capable of being brought 

 to a very significant, although not decisive test, by examining a 

 peculiar and important distribution of the 28 synthemes into 7 

 sets of 4 synthemes each, the property of each set being that its 

 12 triads contain amongst them all the 36 duads appertaining to 

 the 9 elements. I discovered this mode of distribution very 

 many years ago ; but it was first published independently by a 

 mathematician whose name I forget, either in the Philosophical 

 Magazine or in the Cambridge and Dublin Mathematical Journal, 

 I think at some time between the years ] 847-53. A similar 

 mode of distribution exists for any system of elements of which 

 the number is a power of 3. Without pausing to give the law of 

 formation, I shall simply observe that for 9 elements we may 

 take as a basic arrangement the square 



1 2 3 



4 5 6 



7 8 9 

 and form from this, by a symmetrical method, the annexed six 

 derived arrangements : — 



712 723 923 



3 5 6 14 5 15 6 



4 89 689 478 



431 523 4 23 



756 76 4 856 



289 189 197 

 and reading off each of these squares in lines, in columns, and 

 in right and left diagonal fashion, we obtain the 7 sets of 4 syn- 

 themes each referred to, viz. 



123 456 789 

 147 258 369 

 159 267 348 

 168 249 357 



* Communicated by the Author. 



