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



advance to be made than has yet been realized in the theory of 

 the constitution of discriminants) that a cusp may be so modified 

 by the nidus at which it is posited, as, without ever passing into 

 a triple point, to be capable of furnishing any amount of mul- 

 tiplicity whatever, curiously in this contrasting with an ordinary 

 double point, no amount whatever of extraordinary singularity 

 imparted to which, or so to speak, to its nidus, can ever heighten 

 its multiplicity so as to make it surpass the first degree without 

 first converting it into a cusp. I may illustrate the nature of a 

 flex-cusp by what happens to a curve of the third degree. When 

 it breaks up into a line and a right line, there are two ordinary 

 double points ; for the existence of these double points, as for 

 the existence of a cusp, two conditions are required. When, 

 however, the right line and conic touch one another (a casus 

 omissus this in the works of the special geometers), the characters 

 of the cusp and the point of inflexion are combined at the point 

 of contact ; the multiplicity is of the third degree, and the sin- 

 gularity also of a degree not exceeding this ; three conditions 

 only being necessary to be satisfied in order that a given cubic 

 may degenerate into such a form ; and it will be found that the 

 discriminant and the first and second evectants thereof vanish for 

 this case, and that the 3rd evectant of the discriminant will be 

 a perfect 9th power ; whereas in order that the cubic may have 

 a so-called triple point, i. e. may degenerate into a trident of 

 diverging rays, four conditions must be satisfied, and it will be 

 found that when this is the case, the first, second, and third 

 evectants of the discriminant will all vanish, and the fourth will 

 be a perfect 12th power of a linear function of the variables. I 

 may mention, by the way, at this place, that the law of a discri- 

 minant and the successive evectants up to the with inclusive, all 

 vanishing, may be expressed otherwise (not in identical, but in 

 equivalent or equipollent terms), by saying that the discriminant 

 and all its derivatives of a degree not exceeding the mth will all 

 vanish — understanding by a derivative of the discriminant any 

 function obtained from the discriminant by differentiating it any 

 specified number of times with respect to the constants of the 

 function to which it belongs, the same constants being repeated 

 or not indifferently*. And very surprising it must be allowed 

 to be, stated as a bare analytical fact, that (m + 1) conditions 

 imposed upon the coefficients of a function of any number of va- 

 riables and of any degree should suffice to make the inordinately 

 greater number of functions which swarm among the derivatives 

 of the mth and inferior degrees of the discriminant each and all 

 simultaneously vanish. 



* Or, to speak more simply, the discriminant and its successive different 

 tials up to the with exclusive must all vanish simultaneously. 



