On the Representation of Imaginary Points by Real Points on a Plane

Extract

I wish to express my great indebtedness to Mr. P. J. Heawood, of Durham University, for his careful and suggestive criticism of the contents of this paper, without, however, making him responsible for any statement in it with which he may disagree.-A. LODGE.

For the sake of those readers who have not seen the first part of this paper, and to whom the whole subject is new, I may state that the whole argument is based on the fact that any imaginary point such as (6 + i, 4 + 3i) has a simple vector x + iy = (6 + i) + i(4 + 3i) = 3 + 5i, and therefore may be vectorially plotted on to, or represented by, the real point [in this case (3, 5)] which has the same vector. This real point is thus the representative of all the imaginary points which have the same vector. The paper deals with the consequent relations between the representatives of conjugate imaginaries, especially when the imaginaries satisfy the equation of some given curve : in this case the two real points representing such a pair of conjugate imaginaries are called images of each other in the curve.