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Published online by Cambridge University Press: 24 October 2008
The circles in a plane can be represented by the points of space of three dimensions in such a way that the points (point circles) of the plane correspond to the points of a quadric Q in space and two orthogonal circles in the plane correspond to two points conjugate with respect to Q. An ∞2 system σ of circles in the plane corresponds to a surface S in space and the reciprocal Sσ′ of circles in the plane closely and reciprocally related to the system σ. So far as I know only trivial examples of such systems σ and σ′ have hitherto been noticed and my object is to direct attention to an illustration drawn from the modern geometry of the triangle which is less simple and perhaps more interesting.
* The equation of a circle contains four coefficients and we take these or linear functions of them as coordinates of the corresponding point in space. The condition for a point circle is the vanishing of a quadratic form Q in the coefficients and two circles are orthogonal if the two sets of coefficients satisfy the conjugacy condition in regard to Q.
† The reader may refer to the treatise of Coolidge.
* We may ignore the case of coincident isogonal conjagates for it mast be possible to make an arbitrary pedal circle the double element. It may be noted that all cnrves of type (I) that pass through the incentre have a doable point there and the same is true for the excentres. Cf. § 6 infra.
† Cf. preceding footnote.