430 Prof. Sylvester on a linear Method of Eliminating between 



[these it will be useful in future to term the equations^ 

 x% jf", 2-'' being the arguments, and F, G, H, etc. the factors 

 of decomposition] for otherwise calling the indices of x, y, z 

 in any original argument a, b, c, their sum or n would be not 

 greater than (ii + 2) — 3, i. e. (« — 1,) which is absurd. 



For the same reasons as in the last case no index of aug- 

 mentation must be made zero: the degree of each will be {n — a) 



, 1 • 1 {n + \)n 



4 (h — /3) + {n — y,) I.e. (2«— 2,)and then- number -^^ — - — ; 



the number of augmentatives will be 3 . ^^ — ^ linearly un- 



involved, each of the degree 2 n — 2, and therefore containing 



(2?z— 1)2m 



^^ arguments. 



(n+ 1)« , 3.(n- l)n (2 ra - 1) 2 ?t 



Now ^—2 + 2 == 2 ^• 



Hence the final derivee may be found, and it will be in its 



. 3.(n — 1)« , ^ 

 lotioest terms, for every member will contain ^ letters 



due to the augmentative, and — ^ — ^ — '— due to the partial 



derivative equations ; in all then there will be 3 «^ letters in 

 each term. 



This second method being applied to three quadratic equa- 

 tions of the most general form, leads to the problem of elimi- 

 nating between six simple equations which lies within the 

 limits of practical feasibility, and it is my intention to register 

 the final derivee upon the pages of some one of our scientific 

 Transactions as a standing monument for the guidance of 

 hereafter coming explorers*. 



Scholium to Case (A). 

 If we attempt to carry forward these processes to quater- 

 nary systems, it becomes necessary to make 



a + g + r+8=(^- 2)" + U,wherer isthenumber 

 or else a + /3 + y+8=(r — 2)«4-2J' 



of proposees. 



Now if the factors in the equations of decomposition are all 

 integer, one of the indices of derivation must be not greater 

 than the corresponding index in any of the original argu- 

 ments, which may easily be shown to be always impossible for 

 a system of equations, cowjo/e/^ in all their terms, whenever their 



* Elimination between two quadratics leads to a final derivee made up 

 o{ seven terms only; the final derivee of three quadratics is made up of at 

 least several thousand; nay, I believe I may safely say, several myriads of 

 terms ! 



