

Mr. J. J. Sylvester on a new Theory of Multiplicity . 467 



for the precious and pregnant observation on the form assumed 

 by the first discriminantal evectant of a binary function with 

 a pair of equal roots, out of which, combined with some an- 

 tecedent reflections of my own, this new theory of multiplicity 

 has taken its rise. The idea of the process of evection, and the 

 discovery of its fundamental property of generating what, in my 

 calculus of forms (Camb. and Dub. Math. Journ.), I have called 

 contravariants, is due to my friend M. Hermite. The polar 

 reciprocals of curves and other loci are contravariants and, as I 

 have recently succeeded in showing, for curves at least, evectants 

 but of course not discriminantal evectants; and I am already 

 able to give the actual explicit rule for the formation of the polar 

 reciprocal of curves as high as the 5th degree, which with a little 

 labour and consideration can be carried on to the 6th, and in 

 fact to curves of any degree (n) when once we are acquainted 

 with any mode of determining all such independent invariants 

 of a function of two variables as are of dimensions not exceeding 

 2(n — 1) in respect of the coefficients. 



By the special geometers (by whom I mean those who, unvi- 

 sited by a higher inspiration, continue to regard and to cultivate 

 geometry as the science of mere sensible space) this problem has 

 only been accomplished, and that but recently, for curves whose 

 degrees do not exceed the 4th. Mr. Salmon has made the happy 

 and brilliant (and by the calculus of forms instantaneously 

 demonstrable) discovery, communicated to me in the course of a 

 most instructive and suggestive correspondence, that a certain 

 readily ascertainable evectant of every discriminant of any func- 

 tion whatever is an exact power of its polar reciprocal*. 



I believe that it may be shown, that, with the sole exception of 

 odd-degreed functions of two variables, the polar reciprocal itself 

 (as distinguished from a power thereof) of every function is an 

 evectant, not (of course) of the discriminant, but of some deter- 

 minable inferior invariant. 



26 Lincoln's-Inn-Fields, 

 May 14, 1852. 



P.S. The terms pluri-simultaneous and pluri-simultaneity, 

 used or suggested by me in my last paper in the Magazine, may 

 be advantageously replaced by the more euphonious and regu- 

 larly formed words consimultaneous, consimultaneity. Multi- 

 plicity and all its attributes and consequences are included as 

 particular cases in the general conception and theory of consi- 

 multaneity, i. e. of consimultaneous equations, or, which is the 

 same thing, of consimulevanescent functions. 



* Viz. for h function of degree n, and variability (i. e. having a number 

 of variables) p, the (n — l) p-1 th evect of the discriminant is the (»— l)th 

 power of the polar reciprocal. 



2H2 



