Mr. J. J. Sylvester on Aronhold^s Invariants, 371 



tunately threw aside, and which M. Hermite has since demon- 

 strated, that there are more than five Hnearly independent 

 invariants to [x, yY, and more than ten, in fact twelve at least, 

 to (a?, yy^ ; that is to say, it is impossible in the latter case to 

 find ten of which all the rest shall be rational functions, although 

 an algebraical equation connects any 11. So, again, if we take 

 a system of two cubic equations, there are only five absolutely 

 independent invariants ; but there are not less than seven linearly 

 independent fundamental invariants, of which any other invariant 

 fljP^J tl^j^ r^iojaal function. In fact, if we take for our two (^uhjp^ 



T him ^ ^^utti A xj = fl<x?3 + ^bx^y + Scxy^ + df ^i^ *^s 



V=a^3 + 3/3A + 37V + V, '^^^'.^Z.^^ 

 the (5) coefficients of the powers of \ in the discriminant of 

 U + XV, each of which is of four dimensions in the two sets of 

 coefficients combined, are all invariants of the system ; but there 

 will be besides two more, one of which is a Combinant of six 

 dimensions, being the resultant of U and V; the other is a Com- 

 binant of two dimensions only, viz. ffS — 3Z»7 + 3c/3— ^a. These 

 seven together form the fundamental constituent scale. 



The two last-mentioned may be expressed algebraically (by the 

 introduction of square roots) as functions of the other five, but of 

 course not as rational functions of the same. My attention was 

 more particularly called to the search of a proof of the complete- 

 ness of the Aronholdian system of invariants, by an inquiry as to 

 the possibility of rigidly demonstrating that there could exist 

 no others not made up of these, addressed to me in the spring of 

 last year by one of the most gifted geometers of this or any other 

 country. A morning or two after the inquiry reached me, in a 

 walk before breakfast by the side of the ornamental water in 

 St. James's Park (a time and place by no means, according to my 

 experience, unfavourable to the inspirations of the Analytic muse), 

 I had the satisfaction of falling upon the rather joi^wflw^ demon- 

 stration above given, which essentially rests upon a principle, 

 requiring no harder exercise of faith than the concession of the 

 impossibility of a greater being contained in or proceeding out 

 of a less. "J 



7 New Square, Liucoln's Inn, ■ ' 



March 1853. ^^*- 



■n\] 



Erratum, 



In the first part of this paper given last month there is an 

 error of calculation (not, however, aff"ecting the result of the 

 reasoning) in the last paragraph of page 302. The cases of 



