318 ON THE GEOMETRICAL REPRESENTATION OF IMAGINARY VARIABLES [689 
4. Consider, as before, P as describing a closed curve which does not include 
within it any point V, and the corresponding points P' as describing each of them 
a closed curve. As the curve described by P approaches a point V, the curves 
described by two of the points P' will approach the corresponding point (W'); and 
when the curve described by P passes through V, the curves described by the two 
points P' will unite together at this point (W') as a node; viz. the}’ will form a 
figure of eight*, the crossing being at the cross-point (W') } which corresponds to the 
branch-point V. And, corresponding to the closed curve described by P, we have 
this figure of eight (replacing tw’o of the original n closed curves), and n — 2 closed 
curves described by the other points P'. 
5. Supposing, next, that the closed curve described by P (instead of passing- 
through the point V) includes within it the point V, then the figure of eight 
transforms itself into a twice-indented oval*. There are on this curve two of the 
points P' w r hich correspond to the given point P; and as P, moving continuously 
in its closed curve, returns to its original position, the first of these points P', 
moving continuously along a portion of the curve, comes to coincide with the original 
position of the second point P'; while the second point P', moving continuously along 
the remaining portion of the curve, comes to coincide with the original position of 
the first point P'; viz. the two portions of the curve are described by the two points 
P' respectively. The curve may thus be regarded as a bifid curve, belonging to these 
two points P'. And, corresponding to the closed curve described by P, we have this 
bifid curve belonging to the two points P', and n — 2 single closed curves belonging 
to the other n — 2 points P' respectively. 
6. If the closed curve described by P (including within it a point V) comes 
to pass through a second point V, the effect will be a new node at the corre 
sponding point (IF); viz. at this point (IF) either the bifid curve unites itself 
with one of the single curves, or two of the single curves unite together, or the 
bifid curve there cuts itself. And, if the curve described by P comes to include 
within it this second point V, then in the three cases respectively:—the bifid curve 
takes to itself the single curve, so that the system then is a trifid curve and n — S 
single curves; or the two single curves give rise to a bifid curve, so that the 
system is two bifid curves and n — 4 single curves; or, lastly, the bifid curve breaks 
up into two single curves, so that the system resumes its original form of n single 
curves. 
7. We thus see how the closed curve described by P, including within it 
certain of the points V, may be such as to have corresponding to it an a-fid curve, 
a /3-fid curve, &c., (a + /3 + ... = w); viz. an a-fid curve contains upon it a of the 
points P' which correspond to the original position of P; and then, as P describes 
* The name figure of eight refers to the case where the two curves which come to unite at (IF') are 
proper ovals (non-autotomic closed curves). They might have one or both of them a node or nodes, as 
explained in No. 3; and the term would then be inappropriate. And so, lower down, the name twice- 
indented oval is used to express the form into which a proper figure of eight is changed by the disappearance 
of the node.