Pansymmetrical Pencils

Extract

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 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.