54 Prof. Sylvester on Tactic. 



i. e. 13,440, 26,880, 120,960, 120,960, 241,920, 1,088,640 indi- 

 vidual groupings. I conclude with putting a grand question, 

 more easy to propose than to answer, viz. are these one million six 

 hundred thousand (and upwards) groupings (classifiable under 

 six distinct genera) all the possible modes and types of grouping 

 which will satisfy the conditions of the question ? and if not, what 

 other mode or type of grouping can be found ? Were I com- 

 pelled to give an answer to this question, I would say that the 

 balance of my mind leans to the opinion that the six types 

 in question are the sole possible types of solution ; but I do 

 not pretend to rest this judgment upon any solid grounds of 

 demonstration, nor to entertain it with any strong degree of 

 assurance. It is a question which the effort to resolve cannot 

 but react powerfully on our knowledge of the principles of tactic 

 in general, and of the theory of substitution -groups in particular; 

 and as such I submit it to the consideration of the rising chivalry 

 of analysis, seeking myself meanwhile fresh fields and pastures 

 new of meditation. 



K, Woolwich Common, 

 June 6, 1861. 



family as coincident with type : species is the proper term.) The type of 

 a total grouping in the problem referred to in the text will depend not only 

 on the particular combination of the types of the binomial and trinomial 

 partial groupings which give rise to these 6(=2x3) genera, but also on 

 the relative phases of the types so combined. The number of groupings 

 in one type or species is always a submultiple of the number of per- 

 mutations of the elements ; whereas it will be seen that the number of 

 groupings in one of the above genera greatly exceeds that number, which 

 in the present case is only 



1.2.3.4.5.6.7.8.9, or 362,880. 

 Whatever may be the case in natural history, the nature of a type or species, 

 as distinguished from a genus, family, or any other higher kind of aggrega- 

 tion of individuals, in pure syntax is perfectly clear and unambiguous ; 

 those groupings form a species which are commutable into one another by 

 an interchange of elements : thus the different phases of the same type or 

 species are in analogy with the different values of the same function arising 

 out of a change in a constant parameter. If it should turn out that the 

 above sixteen hundred thousand and odd groupings are not the sole solu- 

 tions of which the question admits, then it will follow that even in this 

 early instance we shall have an example not only of species and genera, but 

 of distinct families of genera, for it is certain that the above six genera 

 constitute within themselves a complete natural family. It will form an 

 interesting subject of inquiry to ascertain how many types are included 

 within each of the six genera belonging to this family ; and be it never 

 forgotten that to each species corresponds, and from it is, so to say, capable 

 of being extracted or sublimated, a Cauchian substitution-group. 



