IV. On the equations of circles. (Second memoir.)
In the year 1866 was published in the “Proceedings of the Royal Irish Academy” a paper "On the Equations of Circles, which contained extension of many known theorems. Thus it was proved in it that the same forms of equation which are true for a circle inscribed in a plane or spherical triangle hold also when the right lines in the one case, or the great circles in the other, are replaced by any three circles in the plane or sphere, and it was shown that the transformed equations represented the pairs of circles which touch the three given circles. The results for circles on the sphere were still further, extended, namely, to conics having double contact with a given conic. The paper contained, in addition to these fundamental investigations, many collateral ones on allied subjects. The memoir, of which I now give an abstract, extends the results of the foregoing paper to a polygon of any number of sides inscribed or circumscribed to a given circle. It is proved for the case of circumscribed figures, that the sides of the polygon may be replaced both on the plane and sphere, by circles touching the given circle; and again, these results may be still further extended to conics having double contact with a given conic. A very large amount of geometry is embraced in the paper, and many subjects of much interest are discussed, showing the great fertility of the methods of investigation employed.