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viii

PREFACE.

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 Hart’s 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 ^[1]

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

↑ In this connexion a remark, which it is now too late to insert in its natural place in the text, may be made here.
Just as the constant of Art. 169 is called the Spherical Power of the point with respect to the small circle, so the constant of Art. 171 may be called the Spherical Power of the great circle with respect to the small circle. (If the great and the small circle intersect at an angle ¢, the spherical power is equal to $\tan ^{2}{\frac {1}{2}}\phi$ .) Then, as the radical circle of two small circles is the locus of points whose spherical powers with respect to them are equal, the centre of similitude of two small circles is the envelope of great circles whose spherical powers with respect to them are equal. Of course by the centre of similitude of two circles is meant the external or the internal centre of similitude, according as the circles have the same or opposite senses of rotation assigned to them. This view of centres of similitude completes the analogy between coaxal and colunar circles, whether on a sphere or on a plane.

[1] In this connexion a remark, which it is now too late to insert in its natural place in the text, may be made here.
Just as the constant of Art. 169 is called the Spherical Power of the point with respect to the small circle, so the constant of Art. 171 may be called the Spherical Power of the great circle with respect to the small circle. (If the great and the small circle intersect at an angle ¢, the spherical power is equal to $\tan ^{2}{\frac {1}{2}}\phi$ .) Then, as the radical circle of two small circles is the locus of points whose spherical powers with respect to them are equal, the centre of similitude of two small circles is the envelope of great circles whose spherical powers with respect to them are equal. Of course by the centre of similitude of two circles is meant the external or the internal centre of similitude, according as the circles have the same or opposite senses of rotation assigned to them. This view of centres of similitude completes the analogy between coaxal and colunar circles, whether on a sphere or on a plane.

[1]