ON THE GEOMETRICAL REPRESENTATION OF IMAGINARY 
VARIABLES BY A REAL CORRESPONDENCE OF TWO PLANES. 
[From the Proceedings of the London Mcithematiccd Society, vol. ix. (1878), pp. 31—3.9. 
Read December 13, 1877.] 
In my recently published paper, “ Geometrical Illustration of a Theorem relating 
to an Irrational Function of an Imaginary Variable,” Proceedings of the London 
Mathematical Society, t. vm. (1877), pp. 212—214, [627], I remark as follows:—“If 
we have v a function of u determined by an equation f(u, v) = 0, then to any given 
imaginary value x + iy of u there belong two or more values, in general imaginary, 
of v; and for the complete understanding of the relation between the two imaginary 
variables we require to know the series of values x +iy' which correspond to a given 
series of values x + iy of v, u respectively. We must, for this purpose, take x, y as 
the coordinates of a point P in a plane II, and x, y' as the coordinates of a 
corresponding point P' in another plane IT ”;—and I then proceed to consider the 
particular case where the equation between u, v is u- + v 2 = a 2 , that is, where 
(x + iy) 2 + (x' + iy') 2 = a 2 . 
The general case is that of an equation (*) (u, \) m (v, l) ,l = 0, where to each 
given value, real or imaginary, of u, there correspond n real or imaginary values of 
v; and to each given value, real or imaginary, of v, there correspond to real or 
imaginary values of u. And then, writing u = x + iy and v = x' + iy, and regarding 
{x, y), {x', y') as the coordinates of the points P, P' in the two planes II, 11' 
respectively, we have a i'eal (to, n) correspondence between the two planes; viz. to 
each real point P in the first plane there correspond n real points P' in the second 
plane, and to each real point P' in the second plane there correspond to real points 
P in the first plane. But such real correspondence of two planes does not of 
necessity arise from an equation between the two imaginary variables u, v; and the