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



steps of multiplicity to interpolate between this case and the case 

 analogous (sub modo) to the Ilex-cusp, last considered. Whether 

 these intervening degrees correspond to singularities of an un- 

 ambiguous kind, no one is at present in a condition to offer an 

 opinion. I will conclude with a remark, the result of my expe- 

 rience in this kind of inquiry as far as I have yet gone in it, viz. 

 that it would be most erroneous to regard it as a branch of iso- 

 lated and merely curious or fantastic speculation. Every singu- 

 larity in a locus corresponds to the imposition of certain con- 

 ditions upon the form of its characteristic ; by aid of the theory 

 of evection we are able to connect the existence of these conditions 

 with certain consequences happening to the form of the discrimi- 

 nant, and thereby it becomes possible, upon known principles of 

 analysis, to infer particulars relating to the constitution of the 

 discriminant itself in its absolutely general form, very much upon 

 the same principle as when the values of a function for particular 

 values of its variable or variables are known, the general form of 

 the function thereby itself, to some corresponding extent, becomes 

 known. Thus, for instance, I have by the theory of evection in 

 its most simple application, been led to a representation of the 

 discriminant of a function of two variables under a form very 

 different and very much more complete and fecund in conse- 

 quences than has ever been supposed, or than I had myself pre- 

 viously imagined to be possible. 



According to the opinion expressed by an analyst of the 

 French school, of pre-eminent force and sagacity, it is through 

 this theory of multiplicity, here for the first time indicated, that 

 we may hope to be able to bridge over for the purposes of the 

 highest transcendental analysis, the immense chasm which at pre- 

 sent separates our knowledge of the intimate constitution of 

 functions of two from that of three, or any greater number of 

 variables. 



It is, as I take pleasure in repeating, to a hint from Mr. Cayley*, 

 who habitually discourses pearls and rubies, that I am indebted 



(viz. all those of the cubic coefficient in x, y) over and above what vanish 

 for the case of a so-called triple point is only 4, which is a unit less than 

 the difference between the measures of the multiplicities at the respective 

 points ; and this difference continues to increase as we pass on to so-called 

 quintuple and higher multiple points in the curves. 

 * Mr. Cayley's theorem stood thus : — If 



ax n -\- nbx n ~ l .y + &c. -\-nb .xy"- 1 -{-a'y n 

 have two equal roots, and it be its discriminant, then will 



Jy» * — *»-i.y i*&C. +«!»-!■ "I* 



I da * db — da J 



be a perfect rath power. It will easily be seen that this theorem is con- 

 vertible into a theorem of evection by interchanging in the result x and y 

 with y and — x. 



