Mr. Sylvester on Combinatorial Aggregation. 285 



Formerly I often wished to be personally acquainted with 

 Dr. Barry, in order to reconcile our discordant opinions by 

 a reference to conjoint observations, but now his fancy has 

 so completely got the better of his judgement, and the manner 

 in which he has treated me is so personal and unworthy, that 

 I altogether despair of coming to any understanding with him. 

 I see well that time must decide between him and me, and 

 this, on account of the difficulty of the object, may yet be long 

 in coming about. 



Th. Bischoff, M.D., 



Giessen,Feb. 10, 1844. Professor of Physiology. 



XLIV. Elementary Researches in the Analysis of Combina- 

 torial Aggregation. By J. J. Sylvester, Esq., A.M., 

 F.R.S.* 



I^HE ensuing inquiries will be found to relate to combina- 

 tion-systems, that is, to combinations viewed in an ag- 

 gregative capacity, whose species being given, we shall have to 

 discover rules for ranging or evolving them in classes ame- 

 nable to certain prescribed conditions. The question of nume- 

 rical amount will only appear incidentally, and never be made 

 the primary object of investigation f- 



The number of things combined will be termed the modulus 

 of the system to which they belong. The elements taken 

 singly, or combined in twos, threes, &c, will be denominated 

 accordingly the monadic, duadic, triadic elements, or simply 

 the monads, duads, or triads of the system. 



Let us agree to denote by the word syntheme J any aggre- 

 gate of combinations in which all the monads of a given system 

 appear once, and once only. 



It is manifest that many such synthemes totally diverse in 

 every term may be obtained for a given system to any modulus, 

 and for any order of combination. 



Let us begin with considering the case of duad synthemes. 

 Take the modulus 4- and call the elements a, b, c, d. 



(a .b c.d) i {a .c b . d), (a . d c .b) constitute three perfectly 

 independent synthemes, and these three synthemes include 

 between them all the duad elements, so that no more inde- 



* Communicated by the Author. 



t The present theory may be considered as belonging to a part of ma- 

 thematics which bears to the combinatorial analysis much the same rela- 

 tion as the geometry of position to that of measure, or the theory of num- 

 bers to compilative arithmetic; number, place, and combination (as it 

 seems to the author of this paper) being the three intersecting but distinct 

 spheres of thought to which all mathematical ideas admit of being referred. 



X From avv and -riOripi. 



