By L. Bostock, F.S. Chandler

ISBN-10: 0859500195

ISBN-13: 9780859500197

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**Example text**

To itself (a permutation of the first η integers), for Ρ to be symmetric we require that λ x 2 3 l s x a 2 2 x 2 2 2 φ(2)> · · · - * « ( » ) ) = ^ ( * 1 > * 2 > · · · > * « ) • say that x\ + x\ is a symmetric polynomial in two variables, whereas x\ - x\ is not. Certain important symmetric polynomials in η variables arise as the coefficients of the powers of χ in the expansion of TJ"_ ,(x - x ). Writing this expansion as a polynomial in x, η ^(•*«,(L)> χ S O W E W O U L D t Il(xi-L Xj) =x n - Σ , * " " 1 + a x"- 2 2 +( - \ ) \ , Sec.

0, 0, 1), (0, 0, 0), of which there are a total of 31. T o prove the result for Ν = 0, we observe that in this case the polynomial is a nonzero constant, which can also be considered as a (constant) polynomial in the o,'s. Assuming now that the result is true for Ν < R, we shall prove it for Ν = R + 1. ,", we first observe that /, > i > · · · > /„. For, if there were a j with /'• < i , the permutation which simply interchanges Xj and x would result in a higher term, which, since Ρ is symmetric, must be included in P.

N o w , the construction of Ρ involves a finite n u m b e r of the fundamental constructions, each of which results in a finite n u m b e r of new points as intersections of various types. W e list all such points in order of construction, where points resulting on the same step may be listed in any order. The point Ρ is on this list, say in the M t h position. Omitting anything in the list after Ρ (which could only include points constructed on the same step as P), we have: P „ P , . . , P - , P .

### Applied Mathematics: v. 1 by L. Bostock, F.S. Chandler

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