[Figure 39] FIG. 39 *141. Introduction of infinite point; center of involution.* We connect the projective theory of involution with the metrical, as usual, by the introduction of the elements at infinity. In an involution of points on a line the point which corresponds to the infinitely distant point is called the _center_ of the involution. Since corresponding points in the involution have been shown to be harmonic conjugates with respect to the double points, the center is midway between the double points when they exist. To construct the center (Fig. 39) we draw as usual through _A_ and _A’_ any two rays and cut them by a line parallel to _AA’_ in the points _K_ and _M_. Join these points to _B_ and _B’_, thus determining on _AK_ and _AN_ the points _L_ and _N_. _LN_ meets _AA’_ in the center _O_ of the involution. *142. Fundamental metrical theorem.* From the figure we see that the triangles _OLB’_ and _PLM_ are similar, _P_ being the intersection of KM and LN. Also the triangles _KPN_ and _BON_ are similar. We thus have _OB : PK = ON : PN_ and _OB’ : PM = OL : PL;_ whence _OB · OB’ : PK · PM = ON · OL : PN · PL._ In the same way, from the similar triangles _OAL_ and _PKL_, and also _OA’N_ and _PMN_, we obtain _OA · OA’ : PK · PM = ON · OL : PN · PL,_ and this, with the preceding, gives at once the fundamental theorem, which is sometimes taken also as the definition of involution: _OA · OA’ = OB · OB’ = __constant__,_ or, in words, _The product of the distances from the center to two corresponding points in an involution of points is constant._ *143. Existence of double points.* Clearly, according as the constant is positive or negative the involution will or will not have double points. The constant is the square root of the distance from the center to the double points. If _A_ and _A’_ lie both on the same side of the center, the product _OA · OA’_ is positive; and if they lie on opposite sides, it is negative. Take the case where they both lie on the same side of the center, and take also the pair of corresponding points _BB’_. Then, since _OA · OA’ = OB · OB’_, it cannot happen that _B_ and _B’_ are separated from each other by _A_ and _A’_. This is evident enough if the points are on opposite sides of the center. If the pairs are on the same side of the center, and _B_ lies between _A_ and _A’_, so that _OB_ is greater, say, than _OA_, but less than _OA’_, then, by the equation _OA · OA’ = OB · OB’_, we must have _OB’_ also less than _OA’_ and greater than _OA_. A similar discussion may be made for the case where _A_ and _A’_ lie on opposite sides of _O_. The results may be stated as follows, without any reference to the center: