376 Mr. J. J. Sylvester on u remarkable Theorem in the 



roots will signify a multiplicity 1, two pairs of equal roots, or three 

 equal roots a multiplicity 2 ; a pair of equal roots and a set of three 

 equal roots, a multiplicity 1 + 2 or 3, and so on. Now suppose 

 the total multiplicity of <\> to be m : the first part of the propo- . 

 sitiou consists in the assertion that the 1st, 2nd, 3rd...(m — l)th 

 Evectants of the discriminant of <£, i. e. of the result of elimina- 

 ting x and >/ between -r~, -7- (as well as the discriminant itself), 

 D • dx' chj v 



will all vanish in whatever way the multiplicity is distributed ; 

 the second part of the proposition about to be stated requires 

 that the mode should be taken into account of the manner in 

 which the multiplicity {in) is made up. Suppose, then, that 

 there are (r) groups of roots, for one of which the multiplicity is 

 m u for the second m 2 , &c, and for the rth m C2 , so that »/, +m 2 + 

 ... + ?« 2 = ?«. Then, I say, that the with evectant of the determi- 

 nant of (f> is of the form 



Kr + %)'»!•». {a 2 x + hy) m y n . . . {a q x + h^y) m r - n , 

 where #, : b x r/ 2 :b Q ... a r : b r are the ratios of x : y corresponding 

 to the several sets of equal roots. 



This latter part of the theorem for the case of m = l was dis- 

 covered inductively by Mr. Cayley, by considering the cases 

 when <fi is a function and cubic, or a biquadratic function. I 

 extended the theory to functions of any number of variables, and 

 supplied a demonstration, i. e. for the case of one pair of equal 

 roots. Mr. Salmon showed that my demonstration could be ap- 

 plied to the case of two pairs of equal roots, or two double points, 

 &c v and very nearly at the same time I made the like extension 

 to the case of three equal roots, cusps, &c, and almost imme- 

 diatelyafter I obtained a demonstration for the theorem in its most 

 general form. This demonstration reposes upon a very refined 

 principle, which I had previously discovered but have not yet 

 published, in the Theory of Elimination. 



I have here anticipated a little in speaking of the theorem as 

 applicable to curves and other loci. 



Suppose (f}{x, y, z) = to be the equation to a curve expressed 

 homogeneously. 



Let 



</>(a;, y, z) =ax n +{na'x n - l .y + nb'x"- 1 .z) 



n — l „ .. „ o . / ,m„ „_„ . n—l 



+ n . l_t. a "x»- 2 .y 2 + n{n-l)b"x' l -' 2 .yz + n.—^ -c"*"" 2 .^ 2 , 



-h &c. &c, 

 and understand by the evectant of any quantity the result of 

 operating upon it with the symbol 



xn -ia +* n ~ l -vi +xn -\ z -w +i r"y-.» +&c - 



