426 Prof. Sylvester oti a linear Method of Eliminatmg betweeti 



veloped in a preceding paper, and which has been since 

 adopted and sanctioned by the high authority of M. Cauchy, 

 I call the final derivative : the quantity C is designated the 

 final derivee : and it is our present object to show how this 

 may be obtained in a pritne form, that is to say, divested of 

 irrelevant factors: in this state it must consist of terms, each 

 containing m + n letters, of which n belong to the coefficients 

 of V, and m to those of U. 



Of course in applying this rule it is to be understood that 

 every combination of powers in U or V has a single letter 

 prefixed for its coefficient, and that in the final derivee powers 

 are represented by repetitions of the same character. 



Every term in U or V being of the form C xP . y'^, x^ . y'^ is 

 called an argument, c its prefix. 



Assume two integer positive numbers r and r', and also two 

 others s and s', such that r + r' = ?2 — 1 5 + s' = m — 1, 

 and form from U = V = two new equations, 



^'•./'.U = a;*.y'.V = 0. 

 Such equations are termed the Augmentatives of the two given 

 ones respectively; also^*" .3^'' . U and its fellow are termed the 

 Augmentees of U and V. 



r and r' are termed the indices of augmentation belonging 

 to U, 5 and s' the same belonging to V. 



Finally, it will be useful hereafter to call the given poly- 

 nomials U and V themselves the proposees, and the given 

 equations which assert their nullity, the propositive equations, 

 or, briefly, the propositives. 



Now as many augmentees of either proposee can be formed 

 as there are ways of stowing away between two lockers 

 (vacancies admissible) a number of things equal to the index 

 of the other*; hence we shall have n augmentees of U, and m 

 of V: thus there will be m + n augmentatives each of the 

 degree m + 7i — 1, and the number of arguments is clearly 

 m + n also, so that they can be eliminated linearly, and the 

 final derivee thus found, containing m -\- 11 letters (properly 

 aggregated) in each term, will be in its prime form, that is, 

 incapable of further reduction, and void of irrelevant factors. 



It is worthy of remark, that the final derivee obtained by 

 arranging in square battalion the prefixes of the augmentees, 

 permuting the rows or columns, and reading off diagonal 

 products, affected each with the proper sign (according to the 

 well known rule of Duality), will not only be free from fac- 



♦ " Tot Augnienta utriusvis ex aequationibus propositis formari possunt 

 quot modi sint inter duo receptacula (utrivis vel ambol)Us omiiino vacare 

 licet) rerum, quarum numerus indicem alterius aequat, distributionem faeii 

 endi." 



