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Published online by Cambridge University Press: 24 October 2008
It is well known that in non-euclidean geometry a plane triangle has four circumcircles, and that each of these circles touches each of four other circles. The latter theorem, which is an extension of that of Feuerbach, is essentially due to Hart.
* Proceedings Camb. Phil. Soc. 21 (1923), pp. 297–362.Google Scholar
* The symbol q. signifies with respect to.
* The curve determined by the equation f = 0 will often be referred to as the curve f.
* See (1.21).
* The equation of the order-cubic T 1 may (cf. 2.1) be written in the form
where
and the sides (ξ= 0, η=0, ζ=0, ν=0) of the quadrilateral of reference are respectively B 1C 1, C 1A 1, A 1B 1, A 0B 0C 0. The coordinates (ξ, η, ζ) of the four points IA are then (±α, ±β, ±γ) and the ambiguous signs involved are those referred to in the text; while the forms Ck involve only even powers of the numbers α, β, γ, as is of course implied in the theorem (6.21).
* As in (5.24), C λ, k having double contact with K λ, k (λ = 0, α, β, γ).
† Any plane order-cubic, T 1, of the T-pencil, and the apolar class-cubic, θ1, of the θ-range, may be substituted for T1, θ1, respectively throughout the specification. Since i, k are independent, ∞ 2 systems of sixteen conies [T] may thus be obtained.
* Malgouzou, , Nour: Annales (4), 19 (1919), p. 210.Google Scholar