50 Prof. Sylvester on Tactic. 



Let us form a framework with the nomes 1.2.3, 7.8.9 

 exactly similar to that which we formed before with 4.5.6, 



7.8.9, and let V, V be its two parts respectively analogous to 

 U, IT, we thus obtain for V, 



1.2.7,8.9,3; 7.2.8,7.9,3; 1.2.9,7.8,3 

 2.3.7,8.9,1; 2.3.8,7.9,1; 2.3.9,7.8,1 

 3.1.7,8.9,2; 3.1.8,7.9,2; 3.1.9,7.8,2 



and for V, 



8.9.3,1.2,7; 7.9.3,1.2,8 

 8.9.1,2.3,7; 7.9.1,2.3,8 



7.8.3,1.2,9 

 7.8.1, 2.3,9 

 7.8.2,3.1,9 



8.9.2,3.1,7; 7.9.2,3.1,8; 

 We must now perform the unwonted process of symbolical 

 division, and obtain the quotients of U« by V, and of U«, Vfi, 

 Vy, XJ8, Ue by V (it will of course be perceived that it is known 

 a priori that the dividend forms of arrangement are actual 

 multipliers of the divisors V and V). In writing down the 

 results of these divisions, which will consist exclusively of ele- 

 ments belonging to the nome 4.5.6, and of which each, term 

 will be of the form d, e ./, we may, analogously to what we have 

 done before for greater brevity, write down only the single ele- 

 ment (d), and omit the residue (ef), which is determined when 

 (d) is determined. We shall thus obtain the quotients follow- 

 ing :— 



tt 5 4 6 

 U« =65 4 



v 4 6 5 



Xj a 546 u/3 564 U 7 546 

 ^- = 654 -—=645 ^- = 465 

 V 465 V 456 y 654 



US 5 6 4 tie 4 6 5 



— = 456 -=546 



v 645 v 654 



It may be observed that these divisions may be effected with 

 great rapidity ; because when three out of the nine figures (in 

 any quotient) not in the same line or column are known, all the 



rest are known. Thus, for example, to find _ it is only neces- 

 sary to seek in Ue the syntheme which contains 1.2.7, and 

 then to take out the figure in that syntheme associated with 8 . 9 

 in that line, viz. 4 ; then again to seek the syntheme which con- 



