In a paper under the above title in Vol. XXIV. of the Proceedings it is shown that in a certain system of co-ordinates, the equation of the first degree represents a circle orthogonal to a fixed circle. It follows that any purely graphical theorem regarding right lines in a plane can be extended to orthogonals to a circle. This may be seen otherwise by projecting the figure of right lines on a sphere, the right lines thus becoming circles orthogonal to a circle on the sphere; and then inverting the sphere into the original plane. The geometrical method shows that the extension may also be applied to theorems involving one circle as well as right lines, the circle remaining unchanged, while the lines become orthogonals to a circle; the Pole and Polar Theorem, Pascal's and Brianchon's Theorems are examples. But plane figures involving more than one circle cannot in general be transformed in this way. We cannot, for instance, deduce the construction for a circle touching three great circles on a sphere from the known construction for a circle touching three lines in a plane; nor the Gergonne construction for circles on a sphere from the corresponding method in a plane.