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viii a long time, I have not found any connected account of the subject, such as is contained in Chapter X.

Coaxal circles have been discussed in such a way as to shew their analogy with coaxal circles on a plane; and the coaxal system and the reciprocal of a coaxal system, to which I have given the name colunar, are selected as examples of Duality, partly because the properties of the latter afford a new treatment of Theorem, but chiefly because, on transition to the plane, they present an interesting relation between systems of circles on the plane, possessed in the one case of a common radical axis, in the other of a common centre of similitude

A chapter has been devoted to the generalisation of the Spherical Triangle, based on a recent memoir by Dr ; and another gives a brief account of Prof. application of determinants to the geometry of the sphere.