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V. "On Determinants, better called Eliminants." By Pro- 

 fessor FRANCIS NEWMAN, M.A. Communicated by Dr. 

 BOOTH, F.R.S. &c. Received March 6, 1857. 



(Abstract.) 



1. This paper aimed at recommending the introduction into ele- 

 mentary treatises of the doctrine of Determinants ; which, following 

 Professor Boole, it called Eliminants. It exemplified the great aid 

 to the memory which the notation aifords. It undertook to show, 

 that if only so much of new notation be used, as is needed in ele- 

 mentary applications, the subject becomes full as easy as the second 

 part of algebra. The method of proceeding recommended may be 

 understood by the following concise statement. 



If n linear eqq. are given, connecting n unknown quantities ; and 



every eq. is represented by A^e + B^ 2 + C^x 3 -\ 1- N r ^ Jl = P r (where 



r is 1, 2, 3 ... n in the several eqq.), then, solving for any one of the 

 unknowns, we of course obtain a result of the form mx=a. Very 

 simple considerations then show, that m and a will be integer func- 

 tions of the coefficients : namely, it is easy to prove, that if this is 

 true for one number n, it must needs be true also for the number 

 (n + 1 ) ; and consequently is generally true. Next, the same ana- 

 lysis exhibits, that m=0, is the result obtained, when P P 2 P 3 ... P n all 

 vanish : moreover, that if the system presented for solution be the 

 (w-l)eqq. 



-R 3 v 3 + . . . + B m =0' 



and the solutions are denoted by 



mv=^a l ; wzw 2 =a 2 ; . .. mv n _ 1 ^a n _^ ; 

 we get the relations 



m= A.a l 4- A 2 o 2 + A 3 3 + . . . + A n _ l a n _ r + A n * 



out of which flow all the rules for the genesis of Eliminants, and the 

 application of them to solve linear eqq. of any degree. 



