﻿380 Mr. J. J. Sylvester on two remarkable Resultants arising 



a v a<2, . . . a 2 „ being 2n given unequal quantities. It follows 

 from the above that the number of distinct solutions is 

 l 2 . 3 2 . . . (2n — l) 2 , unless one group of i of the coefficients a and 

 a second group of i other of them can be found such that the sum 

 of the one group is equal to the sum of the other; in that case, 

 and in that case only, the number of solutions undergoes a re- 

 duction. A similar conclusion can be extended to the case of 

 an odd number (2n-\-l) of the parameters (a), in which case 

 the number of solutions is l 2 . 3 2 . . . (2n — l) 2 (2n + l), except 

 when, as above, two sets of parameters can be found the same 

 in number and equal in amount, in which case the number 

 of solutions undergoes a reduction as before. 



I mention these facts with the view of making it understood 

 that the problems of elimination herein proposed and solved are 

 not mere idle dreams and speculations of the fancy, but have a 

 real ontological significance in connexion with a great algebraico- 

 Diophantine problem of the Integral Calculus. 



P.S. Suppose v to be any positive integer, even or odd, and 

 that the curve or compound symmetrical logarithmic wave 



is to be made subject to the relation arc minus abscissa 



Then the a coefficients (or form-parameters) being given, the P 

 quantities (or asymptotic distances from the Y axis of the loga- 

 rithmic wavelets) depend on the solution of an algebraical equa- 

 tion whose degree is the product of v terms of the series 

 1, 1, 3, o, 5, o, /,.... 



When v — 2n, the coefficient of the leading term of this equa- 

 tion is the resultant of the system, or rather double system, of 

 2n functions of 2n variables which has been already discussed. 



"When v = 2/2 + 1, the coefficient of the leading term is the re- 

 sultant of a system of 2n -f 1 functions of 2n -f 1 variables : (n + 1) 

 of them of the form %x, 2^ 3 , . . . 2a? 2n+1 ; n of them of the form 

 %&%, Xax 3 , . . . ^ax 2n+l respectively. 



To obtain this last-named resultant we may pair the variables 

 (leaving one out) in every possible way, then make the sum of 

 each pair and also the solitary or unpaired one zero, and finalty, 

 substituting in the n equations last stated (which come down to 

 the form of a system of n equations between n variables discussed 

 at the outset of this paper) , calculate its resultant *. The product 



* Regarded as loci, the v functions can only interset in one or another 

 of an invariable system of points independent of the particular values of 

 the coefficients. The equations to any one of these points (from what has 



