Prof. Sylvester on the Tactic of 9 Elements. 



145 



712 356 489 



734 



158 



269 



759 



164 238 



768 



139 254 



431 



756 289 



472 358 



169 



459 



362 



178 



468 379 



152 



723 145 689 



716 248 359 



749 



256 318 



758 



219 346 



523 



764 189 



571 



268 349 



569 24 L 378 



548 279 361 



923 156 478 



914 257 368 



988 



264 317 



967 



218 354 



423 



859 197 



481 



259 367 



457 



261 389 



469 287 351 



If, now, we take any distribution of the 9 elements into nomes 

 other than 123, 456, 789, we shall find that some of the syn- 

 themes will contain trinomials, some binomials only, but others 

 (in number either 9 or 18, according to the distribution chosen) 

 will contain binomials and trinomials mixed; but if we adopt 

 123, 456, 789 as the nomes, then it will be found that the 

 remaining 27 synthemes (after excluding the monomial syntheme 

 123, 456, 789) will consist of 18 purely binomial triads, and 9 

 purely trinomial triads. The former will consist of the first, 

 second, and fourth synthemes of the 6 derived groups ; the latter 

 of the second, third, and fourth of the basic group, and of the 

 second synthemes of each of the 6 derived groups. 



It may be remembered that there are two types or species of 

 groupings of trinomial triad synthemes appertaining to 3 nomes of 



3 elements ; one of these species contains 4, the other 36 individual 

 groupings. It may easily be ascertained that the grouping above 

 indicated belongs to the first (the less numerous) of these species. 

 Again, there are 3 types or species of groupings of binomial triad 

 synthemes appertaining to the same system of nomes ; one con- 

 taining 12, one 24, and the third 108 groupings. The group- 

 ing with which we are here concerned will be found to belong to 



the second of these species, — that denoted by the symbols 



in my paper of last month. Hence, then, we derive a very con- 

 siderable presumption in favour of the opinion which I advanced 

 at the close of my preceding paper on Tactic, and derived, too, 

 from a case apparently unfavourable to the verisimilitude of the 

 conjecture ; for a natural subdivision of 28 things into 7 sets of 



4 each seems at first sight hardly compatible with another natural 

 division into 3 sets of 1, 18, and 9 respectively. Notwithstand- 

 ing this seeming incompatibility, we have found that the two 

 methods of decomposition do coexist, owing essentially to the 

 fact that the 7 sets (of 4 synthemes each) stand not in a relation of 



Phil. Mag. S. 4. Vol. 22. No. 145. Aug. 1861. L 



