In Fig. 1, AA1A2A3 … is a rectangular hyperbola, whose asymptotes are OT, OT', and whose vertex is A. We make the following construction

At A1, a point on the hyperbola, a tangent is drawn, cutting OA at P The tangent at A is drawn, cutting OA1 at P1. Let the other tangent from P1 touch the hyperbola at A2. Produce PA1 to cut OA2 at P2. Let the other tangent from P2 touch the hyperbola at A3, and let P1A2 produced cut OA3 at P3.

Consider this process to be continued indefinitely and to be repeated on the other side of the axis OA, A′1 being the point of contact of the other tangent from P We obtain a pencil of rays OAn, OA′n which gradually lie closer together, becoming infinite in number as the asymptotes, which are the ultimate rays of the pencil, are approached. The system is obviously symmetrical about OA, and I shall show that this pencil has further remarkable properties, which I have named “pansymmetry” These properties can be stated thus

If a tangent to the hyperbola at the point An be produced both ways, it will be symmetrically divided by all the rays of the pencil, the p th section on the one side of An being equal to the p th section on the other side.