298 Mr. J. J. Sylvester on the Relation between the 



subject of statement without an enormous periphrasis, and could 

 never have been made the object of distinct contemplation or 

 proof. 



To return to the more immediate object of this communication, 

 suppose that we have any binary function of two sets of quan- 

 tities, x^, Xc^. . . x„ ; li, ^2 • ■ • ^n, of which the general term will 

 be of the form c^, s X *,.-fs ' ficcording to the principles of notation 

 above laid down, nothing can be more natural than to represent 

 Cr.s by the biliteral group ajx.^; the function in question will 

 then take the form 



%a a. .X P ; 



the x's and ^'s denoting quantities, but the a's and a's mere 

 umbrse. The function may then be thrown under the convenient 

 symbolical form 



{a^x^ + a^^+ .. . +a„.xj , , 



So if we confine ourselves to quadratic functions, for which 

 x^, Xc^. . . x^; ^1, ^2 • • • ?« become respectively identical, the 

 general symbolical representation of any such will be 



{a^x^ + a^Xci+ . . . +a^x^)'^. 



The complete determinant will be denoted by 



fffli ffg . . . «„\ 



and any minor determinant of the rth order by 



L«g Oq . . . a^^J 



where 6^, 6^ . . . 6,. are some certain r distinct numbers taken 

 oat of the series 1, 2, 3 ... r. Suppose now that we have 



linearly transformable into 



by means of the («) equations k 'Htj nu 



a2 = ff2*l-.Vl + «2^2-y2+ • • • +«2*„-?/« I. . (E.) 



' ""^' ^„=^,A-yi + ''A-y2+ ••• +"A-y. 



in which equations, be it observed, each coefficient aj)^ is a 

 single quantity, jjerfcctly iudejicndent of the quantities denoted 



