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A dual quadratic transformation associated with the Hessian conics of a pencil
Published online by Cambridge University Press: 31 October 2008
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1. The invariants and covariants of a system of two conics have been much studied2 but little has been said about those of three conies. Three conics 
have a symmetrical invariant Ω123, or in symbolical notation (a b c)2. According to Ciamberlini3 the vanishing of this invariant signifies that the Φ conic of any two of f1, f2, f3 is inpolar with respect to the third; and in a previous paper4 I have derived by symbolical methods a more symmetrical result, viz., if Ω123 vanishes, then u being any line in the plane, u1, u2, u3 are concurrent, where ui is the polar with respect to fi of the pole of u with respect to Φjk.
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- Copyright © Edinburgh Mathematical Society 1940
References
1 The integral 
is a case in point.
2 See Salmon Conic Sections, Ch. xviii, or Sommerville, Analytical Conies, Ch. xx. Taking point-coordinates x, y, z with corresponding line-coordinates l, m, n, a conic 
has a tangential equation 
. Then the vanishing of the invariant 
of the conics 
implies that there are triangles circumscribed to f 1 which are self-polar for f 2, and f 1 is said to be inpolar to f 2. The contravariant conic Φ12 is the envelope of a line whose intersections with f 1 harmonically separate its intersections with f 2.
3 Giorn. di Mat., Napoli, 32 (1886), 141.Google Scholar
4 Proc. Ed. Math. Soc, 2 iv (1935) 258.Google Scholar
1 Sommerville, loc. cit., p. 278Google Scholar; for a fuller discussion of mutual apolarity, see Strazzeri, , Rend. Circ. Mat. di Palermo, 32 (1937), 100.Google Scholar
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