250 Prof. Sylvester ou a neto and more 



None of the parallelogrammatic functions above taken singly, 

 are symmetric functions of the coefficients, but their sum is ; 

 so also is the sum of the product of each into the quantity 



left out. 



Now in general, suppose that the polynomial/a; contains r 

 perfect square factors, so that we have r couples of equal roots 

 belonging to the equation /a: = 0, it is clear that 



QerCr^,^..^) and all the other -L-^i-^^^^-^y- func- 

 tions of which it is the type are severally zero. Moreover, the 

 sum of these or the sum of the products of each by any sym- 

 metrical function of the (;•-!) letters left out will be a sym- 

 metrical function of the coefficients of the powers of x in fx. 

 To express now the affirmative'^ conditions corresponding to 

 the case of there being /• pairs of equal roots, we might em- 

 ploy the r equations. 



But these, except the last, are not the simplest that can be 

 employed ; that is to say, we can write down r others, the 

 terms of which shall be of lower dimensions in respect lo the 

 roots. 



Let/;i denote that any rational symmetrical function of the 

 j«,th degree is to be taken of the quantities which it precedes. 



Then the r equations in question are all contained in the 

 general equation 



S ^/^ (^1 eo ... ^r-i) X (g,.f,+i ...g„ ) 1=0; 



j«, being taken from up to (r — 1) we obtain r equations, which 

 in respect to the roots are respectively of all degrees between 

 n.{n-l)...{n-r + 2) n.jn-l) ... in-r + 2) 



1.2 ... (r-1) 1 .2 ... ir-l) ^ 



reckoned inclusively. 



Now at this stage it is important to remark that the above 

 r equations, although necessary, are not siifficient ; and indeed, 

 no more affirmations of equality can be sufficient to ensure 

 there being r pairs of equal roots. 



• The importance of the restriction hinted at by the use of the word 

 affirmative wiJl appear hereafter. 



