*127. Conics through four points touching a given line.* It is further evident that the involution determined on a line by the system of conics will have a double-point where a conic of the system is tangent to the line. We may therefore infer the theorem _Through four fixed points in the plane two conics or none may be drawn tangent to any given line._ [Figure 35] FIG. 35 *128. Double correspondence.* We have seen that corresponding points in an involution form two projective point-rows superposed on the same straight line. Two projective point-rows superposed on the same straight line are, however, not necessarily in involution, as a simple example will show. Take two lines, _a_ and _a’_, which both revolve about a fixed point _S_ and which always make the same angle with each other (Fig. 35). These lines cut out on any line in the plane which does not pass through _S_ two projective point-rows, which are not, however, in involution unless the angle between the lines is a right angles. For a point _P_ may correspond to a point _P’_, which in turn will correspond to some other point than _P_. The peculiarity of point-rows in involution is that any point will correspond to the same point, in whichever point-row it is considered as belonging. In this case, if a point _P_ corresponds to a point _P’_, then the point _P’_ corresponds back again to the point _P_. The points _P_ and _P’_ are then said to _correspond doubly_. This notion is worthy of further study. [Figure 36] FIG. 36