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: 

Given two pairs of points in an involution of points, if the points of one pair are separated from each other by the points of the other pair, then the involution has no double points.  If the points of one pair are not separated from each other by the points of the other pair, then the involution has two double points.

144. An entirely similar criterion decides whether an involution of rays has or has not double rays, or whether an involution of planes has or has not double planes.

[Figure 40]

FIG. 40

145.  Construction of an involution by means of circles. The equation just derived, OA .  OA’ = OB .  OB’, indicates another simple way in which points of an involution of points may be constructed.  Through A and A’ draw any circle, and draw also any circle through B and B’ to cut the first in the two points G and G’ (Fig. 40).  Then any circle through G and G’ will meet the line in pairs of points in the involution determined by AA’ and BB’.  For