If the point lies on the conic the involution is parabolic, the tangent at the point counting for coincident focal rays.
If the point is within the conic the involution is elliptic, having no focal rays.
It will further be seen that the involution determined by a conic on any line p is a section of the involution, which is determined by the conic at the pole P of p.
§ 83. Foci.—The centre of a pencil in which the conic determines a circular involution is called a “focus” of the conic.
In other words, a focus is such a point that every line through it is perpendicular to its conjugate line. The polar to a focus is called a directrix of the conic.
From the definition it follows that every focus lies on an axis, for the line joining a focus to the centre of the conic is a diameter to which the conjugate lines are perpendicular; and every line joining two foci is an axis, for the perpendiculars to this line through the foci are conjugate to it. These conjugate lines pass through the pole of the line, the pole lies therefore at infinity, and the line is a diameter, hence by the last property an axis.
It follows that all foci lie on one axis, for no line joining a point in one axis to a point in the other can be an axis.
As the conic determines in the pencil which has its centre at a focus a circular involution, no tangents can be drawn from the focus to the conic. Hence each focus lies within a conic; and a directrix does not cut the conic.
Further properties are found by the following considerations:
§ 84. Through a point P one line p can be drawn, which is with regard to a given conic conjugate to a given line q, viz. that line which joins the point P to the pole of the line q. If the line q is made to describe a pencil about a point Q, then the line p will describe a pencil about P. These two pencils will be projective, for the line p passes through the pole of q, and whilst q describes the pencil Q, its pole describes a projective row, and this row is perspective to the pencil P.