Coincidence formulae in geometry

Extract

1. The principal object of this note is to give a new derivation of the coincidence formula in the general correspondence of points in a rational n-space. It is supposed that the correspondence is algebraical and that there are ∞ⁿ pairs of corresponding points P, P′ (in the geometrical sense of ∞ⁿ). Denote by the symbol (i) (n − i)′ the number of pairs P, P′ such that P lies in a general i-space and at the same time P′ in a general (n − i)-space. When (P, P′) tends to a coincidence of the correspondence, the join PP′ tends to some limiting position, which may depend on the way in which (P, P′) tends to the coincidence. In general the limiting lines will lie on a conical figure—a star—with the double point as vertex. In the simplest case of an isolated double point, the star fills the whole n-space. Denote by the symbol ε(i, n) (n − i) the number of times it happens that a double point lies on a general (n − i)-space while the limiting line PP′ meets a general i-space(and lies in the n-space).