In the same manner the tangent at S may be constructed.

62.  Determination of the locus. We now show that it is possible to assign arbitrarily the position of three points, _A__, __B__, and __C__, on the locus (besides the points __S__ and __S’__); but, these three points being chosen, the locus is completely determined._

63. This statement is equivalent to the following: 

Given three pairs of corresponding rays in two projective pencils, it is possible to find a ray of one which corresponds to any ray of the other.

64. We proceed, then, to the solution of the fundamental

PROBLEM:  Given three pairs of rays, _aa’__, __bb’__, and __cc’__, of two protective pencils, __S__ and __S’__, to find the ray __d’__ of __S’__ which corresponds to any ray __d__ of __S__._

[Figure 12]

FIG. 12

Call A the intersection of aa’, B the intersection of bb’, and C the intersection of cc’ (Fig. 12).  Join AB by the line u, and AC by the line u’.  Consider u as a point-row perspective to S, and u’ as a point-row perspective to S’. u and u’ are projectively related to each other, since S and S’ are, by hypothesis, so related.  But their point of intersection A is a self-corresponding point, since a and a’ were supposed to be corresponding rays.  It follows (§ 52) that u and u’ are in perspective position, and that lines through corresponding points all pass through a point M, the center of perspectivity, the position of which will be determined by any two such lines.  But the intersection of a with u and the intersection of c’ with u’ are corresponding points on u and u’, and the line joining them is clearly c itself.  Similarly, b’ joins two corresponding points on u and u’, and so the center M of perspectivity of u and u’ is the intersection of c and b’.  To find d’ in S’ corresponding to a given line d of S we note the point L where d meets u.  Join L to M and get the point N where this line meets u’. L and N are corresponding points on u and u’, and d’ must therefore pass through N.  The intersection P of d and d’ is thus another point on the locus.  In the same manner any number of other points may be obtained.

65. The lines u and u’ might have been drawn in any direction through A (avoiding, of course, the line a for u and the line a’ for u’), and the center of perspectivity M would be easily obtainable; but the above construction furnishes a simple and instructive figure.  An equally simple one is obtained by taking a’ for u and a for u’.