Similarly the correspondence between two reciprocal pencils is determined if for four rays in the one the corresponding planes in the other are given.
§ 93. We may now combine—
1. Two reciprocal pencils.
Each ray cuts its corresponding plane in a point, the locus of these points is a quadric surface.
2. Two projective pencils.
Each plane cuts its corresponding plane in a line, but a ray as a rule does not cut its corresponding ray. The locus of points where a ray cuts its corresponding ray is a twisted cubic. The lines where a plane cuts its corresponding plane are secants.
3. Three projective pencils.
The locus of intersection of corresponding planes is a cubic surface.
Of these we consider only the first two cases.
§ 94. If two pencils are reciprocal, then to a plane in either corresponds a line in the other, to a flat pencil an axial pencil, and so on. Every line cuts its corresponding plane in a point. If S1 and S2 be the centres of the two pencils, and P be a point where a line a1 in the first cuts its corresponding plane α2, then the line b2 in the pencil S2 which passes through P will meet its corresponding plane β1 in P. For b2 is a line in the plane α2. The corresponding plane β1 must therefore pass through the line a1, hence through P.