Prof. Sylvester on Tactic. 47 



I shall understand the complementary part of the framework, 

 viz. : — 



8.9.6,4.5,7; 7.9.6,4.5,8; 7.8.6,4.5,9 

 8.9.4,5.6,7; 7.9.4,5.6,8; 7.8.4,5.6,9 

 8.9.5,6.4,7; 7.9.5,6.4,8; 7.8.5,6.4,9 



It is of cardinal importance to notice that the order in which 



the imperfect synthemes are arranged in U and U is one of ab- 

 solute reciprocity. It is in this reciprocity, and in the fact of 

 U or U being each in strict regimen (so to say) with the other, 

 that the cause of the success of the method about to be applied 

 essentially resides. 



The slightest reflection will serve to show that every complete 

 syntheme of the kind required will be of the form 



I UxP I 

 I UxP I 



where the symbolical multipliers P and P are each of them some 

 one of the forms (by no means necessarily the same) represented 

 generally by the framework of defective synthemes hereunder 

 written (defective in the sense that all the elements of the 

 second and third nomes are supposed to be omitted), 



, a, be; , b, ca ; 

 , b, ca; , c, ab; 

 , c, ab; , a, be ; 



, c, ab 

 , a, be 

 , b, ca 



or else by the cognate framework 





, a, be ; , c, ab; 

 , b, ca; , a, be; 

 , c, ab; , b, ca; 



, b, ca 

 c, ab 

 a, be 



where a, b, c are identical in some order or another with the 

 elements of the first nome, viz. 1, 2, 3; so that there are six 

 different systems of a, b, c in each of these two frameworks. 



No other combination of the elements in U or U (all of which 

 belong to the second and third nomes) with the elements in the 

 first nome is possible ; for any such combination would involve 

 the fact of a repetition of the same triad or triads in the same 

 grouping, contrary to the nature of a grouping. Hence, then, 

 the number of forms of P and of P being twice six, or 12, we at 

 once perceive that the total number of groupings is 12 x 12, 

 or 144. 



But now comes the more difficult question of ascertaining 

 between how many distinct species or types these groupings are 

 distributed. If we study the form of P or P, it is obvious that 



