370 Mr. J. J. Sylvester on Aronhold's Invariants, 



Hants to the cubic ternary form, entering as the exclusive ingre- 

 dients into every other invariant that can be derived from such 

 form. i'iiiiw 



A word of warning is necessary before I lay down my perfy 

 that there can be only two algebraically independent invariant* 

 to {x, tjY or {Xy y, r)^, is an immediate consequence of the ca- 

 nonical form of each having but one parameter; so in general there 

 can be at most but (n— 2) absolutely independent invariants of 

 [Xy yY ; but the point established in the preceding investigation 

 goes to show that there can exist no other invariants than such 

 as are i-ational functions of s and t in the one case, and S and T 

 in the other. I shall take some other occasion to establish a 

 similar conclusion for the forms {x, yY and [x, y)^. 



I have shown that there exist three invariants to the one of the 

 degrees 4, 8, 12, and four to the other of the degrees 2, 4, 6, 10 ; 

 and I shall demonstrate that any other invariant to either form 

 must be a rational function of those above stated. For the cubic 

 form {Xy yY we know that there is but one invariant, viz. its dis- 

 criminant. Thus, then, for w = 3, n = 4, w = 5, n = 6 the number 

 of absolutely independent invariants is n— 2, and the number of 

 linearly independent invariants is no greater. But this result is 

 by no means generally true. It may be proved by means of a 

 great law of reciprocity* which I myself originated, but unfocr 



* The theorem of reciprocity alluded to in the text is the following : — 

 If to any function [x, y)" there exists an invariant of the order m in the 

 coefficients, then to {x, y)"* there exists an invariant of the order («) in the 

 coefficients ; or more generally, which is M. Hermite's addition, if to any 

 system of functions {x, y)"i, (a?, y)"2, . . . (a?, y)"* there exists an invariant of 

 the several dimensions mj, mg, . . . m^ in the respective sets of coefficients, 



then conversely to a system {x, y)*"i, (a?, y)"*3, ... (a?, y)"*' there exists ao 

 invariant of the dimensions Wj, ng, . . . n^ in the respective sets of coefficients. 

 I had previously shown in this Magazine that Mr. Cayley's formulae 

 for finding the number of biquadratic invariants to any function (a?, y)**, 

 given in that remarkable paper of his on linear transformations in the Cam- 

 bridge and Dublin Mathematical Journal, where first dawned upon the world 

 the clear and full-formed idea of invariants (the most original and important 

 infused into analysis since the discovery of fluxions), could be expressed by 

 means of the number of solutions of the equation in integers 2a7+3y=», 

 the square of the quadratic invariant (which only exists for even values of w) 

 counting for one in the fundamental biquadratic scale ; this is of course a 

 direct consequence, through the law of reciprocity, of the fundamental 

 scale to {x, y)* consisting of a quadratic and a cubic invariant. My dis- 

 covery of the fundamental scale of invariants to (a?, yf and (a?, yY now 

 enables us, through the same law of reciprocity, to express the number of 

 distinct Quintic and Sextic invariants to (x, y)», viz. as being the number of 



integer solutions of a?-|-2y-f 3z=- in the one cascandof a?-|-2y-)-30+6<==- 

 in the other. 



