286 Mr. Sylvester's Elementary Researches in the Analysts of 



pendent synthemes can be obtained from them. Again, let 

 a, b, c, d, e,f be the monads ; we can write down five inde- 

 pendent synthemes, to wit, 



a . b c .d e .f- 



a .d c .f e . b 



a.c d. e f.b 



a .f b .d c . e 



a . e d .f b .c. 



We can write no more than these without repeating duads 

 which have already appeared*. 



We propose to ourselves this problem : — A system to any 

 even\ modulus being given to arrange the "whole of' its duads % in 

 the form of synthemes; or in other words, to evolve a Total 

 of duad synthemes to any given even modidus§. 



When the modulus is odd, as before remarked, the forma- 

 tion of a duad syntheme is of course impossible, for any num- 

 ber of duads must necessarily contain an even number of mo- 

 nadic elements; but there is nothing to prevent us from form- 

 ing in all cases what may be termed a bisyntheme or diplo- 

 theme, i. e. an aggregate of combinations, where each element 

 occurs twice and no more. 



* Such an aggregate of synthemes may be therefore termed a Total. 



f Tiie modulus must be even, as otherwise it is manifest no single syn- 

 theme can be formed. We shall before long extend the scope of our in- 

 quiry so as to take in the case of odd moduli. 



X Triadic systems will be treated of hereafter. 



§ It is scarcely necessary to advert here to the fact of the problem being 

 in general indeterminate and admitting of a great variety of solutions. 



When the modulus is four there is only one synthematic arrangement 

 possible, and there is no indeterminateness of any kind ; from this we can 

 infer, a priori, the reducibility of a biquadratic equation; for using <p,f, F 

 to denote rational symmetrical forms of function, it follows that 



if(<p'a7b,<Pc7dU . 

 , 7 — ; , is itself a rational symmetric function 

 /OPf^.flMn £a,b,e,d. 

 f((pa,d,(pb,c)} 

 Whence it follows that if a,b,c, d be the roots of a biquadratic equation, 

 f((pa,b,<pcd) can be found by the solution of a cubic: for instance, 

 (a + b) x (c-\-d) can be thus determined, whence immediately the sum of 

 any two of the roots comes out from a quadratic equation. 



To the modulus 6 there are fifteen different synthemes capable of being 

 constructed j at first sight it might be supposed that these could be classed 

 in natural families of three or of five each, on which supposition the equa- 

 tion of the sixth degree could be depressed; but on inquiry this hope will 

 prove to be futile, not but what natural affinities do exist between the totals ; 

 but in order to separate them into families each will have to be taken twice 

 over, or in other words, the fifteen synthemes to modulus 6 being redupli- 

 cated subdivide into six natural families of five each. Again, it is true that 

 the triads to modulus 6 (just like the duads to modulus 4) admit of being 

 thrown into but one synthematic total, but then this will contain ten syn- 

 themes, a number greater than the modulus itself. 



