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Published online by Cambridge University Press: 24 October 2008
If A0, A1, A2, A3 are four arbitrary circles, it is well known that there are eight circles which cut them all at equal angles, and that these eight circles fall into two tetrads B0, B1, B2, B3, C0, C1, C2, C3 which are in desmic position to each other and to the tetrad of circles orthogonal to threes of A0, A1, A2, A3.
* For a study of the cubic transformations and the cubic complex, see my “Memoir on Cubic Transformations associated with a Desmic System,” Journal of the Indian Math. Soc., Supplement to vol. 17, 1927, where further references will be found.Google Scholar
† See “Memoir on Cubic Transformations, etc.” loc. cit. p. 13.Google Scholar
* The cubic C is of course symmetrically related to the three triangles, and is equally the jumelaire cubic of β0 with respect to β1β2β3 to γ1γ2γ3. From the fact that C is circumscribed to the quadrangle (αr), as well as to its harmonic triangle, we can conclude from the properties of the cubic that the tangents to C at α0, α1, α2, α3 meet at a point α0 on the curve, and that the tangents at α0, α1, α2, α3 also cointersect on the curve. But, to shew that α0 is the circumcentre of α1α2α3, we require the property of Г1 stated.Google Scholar
* This will be evident from the incidence-scheme given by Hudson, : Kummer's Quartic Surface, p. 7.Google Scholar