146 Prof. Sylvester on the Tactic of 9 Elements. 



indifference set to set, but are to be considered as composed of a 

 base and 6 derivatives in differently related to the base and to each 

 other. The theory of these 7 sets is extremely carious, and well 

 worthy of being fully investigated by the student of tactic, but 

 cannot be gone into within the limits suitable to the pages of 

 a philosophical miscellany. 



Before taking final leave of the subject (at all events for the 

 present, and in the pages of this Magazine), as I have been 

 questioned as to the meaning of the important word " syntheme," 

 derived from auv Qr)\xa, I repeat that a "syntheme" is the general 

 name for any consociation of the single or combined elements of 

 a given system of elements in which each element is once and 

 once only contained. A nome, from vefiw (to divide), means a 

 consociation of a certain number out of a given system of ele- 

 ments; and a binomial, trinomial, or r-noraial combination of 

 any specified sort, means a combination whose elements are dis- 

 persed between 2, 3, or r of the nomes between which the entire 

 system of elements is supposed to have been divided. 



K, Woolwich Common, 

 July 14, 1861. 



P.S. I have found the date and place of the resolution into 7 

 sets referred to in the text ; it is given in a paper by Mr. Kirk- 

 man, vol. v. p. 261 of the Cambridge and Dublin Mathematical 

 Journal for 1850. His 7 squares, whose horizontal, vertical, and 

 two diagonal readings (like mine) constitute the 7 sets in ques- 

 tion, are substantially as follows: — 



1 2 3 

 4 5 6 



7 8 9 



1 2 4 



4 5 7 



7 8 1 



5 6 7 



8 9 1 



2 3 4 



8 9 3 



2 3 6 



5 6 9 



7 1 2 



1 4 5 



4 7 8 



3 4 5 



6 7 8 



9 1 2 



6 8 9 



9 2 3 



3 5 6* 



On assuming 123, 456, 789 as the three nomes, the 28 syn. 



* By changing the positions of the lines and columns of the six deriva- 

 tive squares, which may be done without affecting the value of their read- 

 ings, they may be represented under the form following, which will be seen 

 to render much clearer their relation to the primitive square : — 



412 623 



423 



| 239 



129 



127 



756 745 



956 



451 



563 



453 



389 189 



173 



1 786 



784 



896 



