Pliicker sentence example

pliicker
  • In regard to the ordinary singularities, we have m, the order, n „ class, „ number of double points, Cusps, T double tangents, inflections; and this being so, Pliicker's ” six equations ” are n = m (m - I) -2S -3K, = 3m (m - 2) - 6S- 8K, T=Zm(m -2) (m29) - (m2 - m-6) (28-i-3K)- I -25(5-1) +65K-1114 I), m =n(n - I)-2T-3c, K= 3n (n-2) - 6r -8c, = 2n(n-2)(n29) - (n2 - n-6) (2T-{-30-1-2T(T - I) -1-6Tc -}2c (c - I).
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  • It is implied in Pliicker's theorem that, m, n, signifying as above in regard to any curve, then in regard to the reciprocal curve, n, m, will have the same significations, viz.
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  • With regard to the demonstration of Pliicker's equations it is to be remarked that we are not able to write down the equation in point-co-ordinates of a curve of the order m, having the given numbers 6 and of nodes and cusps.
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  • We have thus finally an expression for = m (m-2) (m2-9) - &c.; or dividing the whole by 2, we have the expression for given by the third of Pliicker's equations.
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  • It is obvious that we cannot by consideration of the equation u = o in point-co-ordinates obtain the remaining three of Pliicker's equations; they might be obtained in a precisely analogous manner by means of the equation v= o in line-co-ordinates,but they follow at once from the principle of duality, viz.
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  • To complete Pliicker's theory it is necessary to take account of compound singularities; it might be possible, but it is at any rate difficult, to effect this by considering the curve as in course of description by the point moving along the rotating line; and it seems easier to consider the compound singularity as arising from the variation of an actually described curve with ordinary singularities.
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  • So that, in fact, Pliicker's equations properly understood apply to a curve with any singularities whatever.
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  • By means of Pliicker's equations we may form a table - The table is arranged according to the value of in; and we have m=o, n= r, the point; m =1, n =o, the line; m=2, n=2, the conic; of m = 3, the cubic, there are three cases, the class being 6, 4 or 3, according as the curve is without singularities, or as it has 1 node or r cusp; and so of m =4, the quartic, there are ten cases, where observe that in two of them the class is = 6, - the reduction of class arising from two cusps or else from three nodes.
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  • At any one of the m 2 -26 - 3K points the variable curve and the consecutive curve have tangents distinct from yet infinitesimally near to each other, and each of these two tangents is also infinitesimally near to one of the n 2 -2T-3t common tangents of the two curves; whence, attending only to the variable curve, and considering the consecutive curve as coming into actual coincidence with it, the n 2 -2T-3c common tangents are the tangents to the variable curve at the m 2 -26-3K points respectively, and the envelope is at the same time generated by the m 2 -26-3K points, and enveloped by the n2-2T-3c tangents; we have thus a dual generation of the envelope, which only differs from Pliicker's dual generation, in that in place of a single point and tangent we have the group of m2-26-3K points and n 2 -2T-3c tangents.
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  • The system has singularities, and there exist between m, r, is and the numbers of the several singularities equations analogous to Pliicker's equations for a plane curve.
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