*124. Double-points in an involution.* The points _C_ and _C’_ describe projective point-rows, as may be seen by fixing the points _L_ and _M_. The self-corresponding points, of which there are two or none, are called the _double-points_ in the involution. It is not difficult to see that the double-points in the involution are harmonic conjugates with respect to corresponding points in the involution. For, fixing as before the points _L_ and _M_, let the intersection of the lines _CL_ and _C’M_ be _P_ (Fig. 33). The locus of _P_ is a conic which goes through the double-points, because the point-rows _C_ and _C’_ are projective, and therefore so are the pencils _LC_ and _MC’_ which generate the locus of _P_. Also, when _C_ and _C’_ fall together, the point _P_ coincides with them. Further, the tangents at _L_ and _M_ to this conic described by _P_ are the lines _LB_ and _MB_. For in the pencil at _L_ the ray _LM_ common to the two pencils which generate the conic is the ray _LB’_ and corresponds to the ray _MB_ of _M_, which is therefore the tangent line to the conic at _M_. Similarly for the tangent _LB_ at _L_. _LM_ is therefore the polar of _B_ with respect to this conic, and _B_ and _B’_ are therefore harmonic conjugates with respect to the double-points. The same discussion applies to any other pair of corresponding points in the involution. [Figure 34] FIG. 34 *125. Desargues’s theorem concerning conics through four points.* Let _DD’_ be any pair of points in the involution determined as above, and consider the conic passing through the five points _K_, _L_, _M_, _N_, _D_. We shall use Pascal’s theorem to show that this conic also passes through _D’_. The point _D’_ is determined as follows: Fix _L_ and _M_ as before (Fig. 34) and join _D_ to _L_, giving on _MN_ the point _N’_. Join _N’_ to _B_, giving on _LK_ the point _K’_. Then _MK’_ determines the point _D’_ on the line _AA’_, given by the complete quadrangle _K’_, _L_, _M_, _N’_. Consider the following six points, numbering them in order: _D = 1_, _D’ = 2_, _M = 3_, _N = 4_, _K = 5_, and _L = 6_. We have the following intersections: _B = (12-45)_, _K’ = (23-56)_, _N’ = (34-61)_; and since by construction _B_, _N_, and _K’_ are on a straight line, it follows from the converse of Pascal’s theorem, which is easily established, that the six points are on a conic. We have, then, the beautiful theorem due to Desargues: _The system of conics through four points meets any line in the plane in pairs of points in involution._ *126.* It appears also that the six points in involution determined by the quadrangle through the four fixed points belong also to the same involution with the points cut out by the system of conics, as indeed we might infer from the fact that the three pairs of opposite sides of the quadrangle may be considered as degenerate conics of the system.