689] ON THE GEOMETRICAL REPRESENTATION OF IMAGINARY VARIABLES. 317 
question of the real correspondence of two planes may he considered in itself, without 
any reference to such origin. 
I was under the impression that the theory was a known one; but I have not 
found it anywhere set out in detail. It is to be noticed that, although intimately 
connected with, it is quite distinct from (and seems to me to go beyond) that of a 
Riemann’s surface. Riemann represents the value u, = x + iy, by a point P whose 
coordinates are x, y; but he considers v!, = x' + iy', as a given imaginary value 
attached to the point P, without representing this value by a point P', coordinates 
x', y'. 
I proceed to consider the general theory of the real (m, n) correspondence. 
Points in the first plane are denoted by the unaccented letters P, Q,..; and the 
corresponding points in the second plane are in general denoted by the same letters 
accented; but there are, as will be explained, special points V, W where the letters 
are interchanged; viz. to the points V or W in the first plane correspond points 
W or V' in the second plane. 
1. To a point P there correspond in general n distinct points P'; and as P 
varies continuously, each of the points P' also varies continuously. 
2. There are certain points V called branch-points (Verzweigungspunkte), such 
that to each point V there correspond two united points, represented by (IT'), aud 
n — 2 other distinct points W'. The points (W’) are called cross-points, aud the 
number of them is of course equal to that of the branch-points V. 
It is throughout assumed that a point denoted by a letter other than V is not 
a point V. 
3. If the point P, moving continuously, describe a closed curve so as to return 
to its original position, then, if this curve includes within it no point V (or all the 
points F)*, each of the corresponding points P' will describe continuously a closed 
curve returning into its original position. Supposing that the curve described by P 
is an oval (non-autotomic closed curve), and taking this to be in the first instance 
an indefinitely small oval, then the curves described by the points P' will in the 
first instance be each of them an indefinitely small oval; but it is worth while to 
notice how, as the oval described by P increases, any one of the ovals described by 
a point P' may become autotomic; viz. if the oval described by P passes through 
two points Q, Q of the m points Q which correspond in the first plane to the same 
Q' in the second plane, then Q' will be a node in the closed curve described by 
that point P' which in the course of its motion comes to pass through Q'. This 
curve is in general an inloop curve composed of two loops, one wholly within the 
other (united at the point Q'), and such that they each include one and the same 
point V' (viz. V' is included within the inner loop): as to this, see post, Nos. 9 
and 10. It will be observed that this node Q' is not a point (W') nor any other 
special point of the second plane. 
* The two cases of the closed curve including no point V, and including all the points V, are really 
identical, as the discontinuity at infinity may be disregarded. It is to be observed that, this being so, it 
follows that the number of the points V must be even.