Pansymmetrical Pencils

In Fig. 1, AA 1 A 2 A 3 … is a rectangular hyperbola, whose asymptotes are OT , OT' , and whose vertex is A . We make the following construction At A 1 , a point on the hyperbola, a tangent is drawn, cutting OA at P The tangent at A is drawn, cutting OA 1 at P 1 . Let the other tangent from P 1 touch the hyperbola at A 2 . Produce PA 1 to cut OA 2 at P 2 . Let the other tangent from P 2 touch the hyperbola at A 3 , and let P 1 A 2 produced cut OA 3 at P 3 . 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 OA n , 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 A n 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 A n being equal to the p th section on the other side.