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 [pg 39] 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'.
66. Lines joining four points of the locus to a fifth. Suppose that the points S, S', B, C, and D are fixed, and that four points, A, A1, A2, and A3, are taken on the locus at the intersection with it of any four harmonic rays through B. These four harmonic rays give four harmonic points, L, L1 etc., on the fixed ray SD. These, in turn, project through the fixed point M into four harmonic points, N, N1 etc., on the fixed line DS'. These last four harmonic points give four harmonic rays CA, CA1, CA2, CA3. Therefore the four points A which project to B in four harmonic rays also project to C in four harmonic rays. But C may be any point on the locus, and so we have the very important theorem,
Four points which are on the locus, and which project to a fifth point of the locus in four harmonic rays, project to any point of the locus in four harmonic rays.