Theorem. If a point be taken on the radical axis of a coaxial system of circles, and from it tangents be drawn to any circle of the system, these tangents are cut in points on a conic, by the radical axis of the circle and a given fixed point. The two points are the foci of the conic. (Fig. 1.) Let W₁W₂ be the line ofgives an ellipse as the locus of P₁P₂, &c, when, as in Fig, 1, S is internal to F. If S were an external point, we should have P₁F-P₁S = P₁F - P₁f₁ = radius of F = constant, and the locus of P₁, P₂, &c., would be a hyperbola. When F is at infinity on the radical axis, P₁S = P₁f₁, and P₁f₁ being at right angles to W₁W₂, the conic is a parabola, and the line of centres the directrix.