_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 if such a circle meets the line in the points _CC’_, then, by the theorem in the geometry of the circle which says that _if any chord is __ drawn through a fixed point within a circle, the product of its segments is constant in whatever direction the chord is drawn, and if a secant line be drawn from a fixed point without a circle, the product of the secant and its external segment is constant in whatever direction the secant line is drawn_, we have _OC · OC’ = OG · OG’ =_ constant. So that for all such points _OA · OA’ = OB · OB’ = OC · OC’_. Further, the line _GG’_ meets _AA’_ in the center of the involution. To find the double points, if they exist, we draw a tangent from _O_ to any of the circles through _GG’_. Let _T_ be the point of contact. Then lay off on the line _OA_ a line _OF_ equal to _OT_. Then, since by the above theorem of elementary geometry _OA · OA’ = OT__2__ = OF__2_, we have one double point _F_. The other is at an equal distance on the other side of _O_. This simple and effective method of constructing an involution of points is often taken as the basis for the theory of involution. In projective geometry, however, the circle, which is not a figure that remains unaltered by projection, and is essentially a metrical notion, ought not to be used to build up the purely projective part of the theory. *146.* It ought to be mentioned that the theory of analytic geometry indicates that the circle is a special conic section that happens to pass through two particular imaginary points on the line at infinity, called the _circular points_ and usually denoted by _I_ and _J_. The above method of obtaining a point-row in involution is, then, nothing but a special case of the general theorem of the last chapter (§ 125), which asserted that a system of conics through four points will cut any line in the plane in a point-row in involution. [Figure 41]