46 Prof. Sylvester on Tactic. 



nomes of three elements each, we advance just one step further in 

 the direction of type-complexity J that is to say, we meet with the 

 existence of 3, and not more than 3, types or species in which all 

 such groupings are comprised. The investigation by which this 

 is made out appears to me well worthy to be given to the world, 

 as affording an example of a new and beautiful kind of analysis 

 proper to the study of tactic, and thus lighting the way to the 

 further opening up of this fundamental doctrine of mathematic, 

 the science of necessary relations, of which, combined with logic (if 

 indeed the two be not identical), tactic appears to me to constitute 

 the main stem from which all others, including even arithmetic 

 itself, are derived and secondary branches. The key to success in 

 dealing with the problems of this incipient science (as I suppose of 

 most others) must be sought for in the construction of an apt 

 and expressive notation, and in the discovery of language by 

 force of which the mind may be enabled to lay hold of complex 

 operations and mould them into simple and easily transmissible 

 forms of thought. I must then entreat the indulgence of the 

 reader if, in this early grappling with the difficulties of a new 

 language and a new notation, I may occasionally appear wanting 

 in absolute clearness and fullness of expression. 



Let us, as before, represent the nine elements by the numbers 

 from 1 to 9, and suppose the nomes to be 1, 2, 3 : 4, 5, 6 : 7, 8, 9. 

 If we take any syntheme formed out of the binomial triads 

 belonging to the above nomes, and if out of such syntheme we 

 omit the elements 1, 2, 3 (belonging to the 1st nome) wherever 

 they occur, the slightest consideration will serve to show that 

 the synthemes thus denuded will assume the form l.m . r,p . q, n, 

 where /, m, r may be regarded as belonging to one of the remain- 

 ing nomes, and p, q, n to the other. The total number of syn- 

 themes in a grouping which contains all the binomial triads is 

 18, because the total number of these triads is 54; and conse- 

 quently it will be seen that every grouping will in fact consist of 

 the same framework, so to say, of combinations of elements be- 

 longing to the second and third nomes variously compounded 

 with the elements of the first nome. 



This framework may be advantageously divided into two parts, 

 each containing nine terms, and which I shall call respectively 

 U and U. Thus by U I shall understand the nine arrange- 

 ments following : — 



4.5.7,8.9,6; 4.5.8,7.9,0; 4.5.9,7.8,6 

 5.6.7,8.9,4; 5.6.8,7.9,4; 5.6.9,7.8,4 

 6.4.7,8.9,5; 6.4.8,7.9,5; 6.4.9,7.8,5 

 each imperfect or defective syntheme being separated from the 

 next by a semicolon, or else by a change of line. So by U 



