689] 
BY A BEAL CORRESPONDENCE OF TWO PLANES. 
319 
continuously its closed curve, returning to its original position, each of these points 
P' describes a portion of the a-fid curve, passing from its original position to the 
original position of a point P' next to it upon the a-fid curve; and the like as to 
a /3-fid curve, &c. The numbers a, /3, ... are not of necessity unequal, and we may 
have sets of equal numbers in any manner. It is hardly necessary to remark that, 
if the curve described by P passes through any point or points V, then two of the 
curves described by the points P' will unite together, or it may be that one of these 
will cut itself at the corresponding point or points (W'); and further that, as in 
No. 3, if the curve described by P passes through two or more of the points Q 
which correspond to the same point Q', then any such point Q' will present itself 
as a node upon the curve belonging to some point, or set of points, P'. But the 
order of succession in which the original n single curves unite themselves together 
into multifid curves, or again break up into single curves, cannot, it would appear, 
be explained in any general manner, and would in each case depend on the nature 
of the particular correspondence. 
8. We may consider the case where the closed curve described by P cuts 
itself. The curve may here be considered as made up of two or more ovals, or, to 
use a more appropriate term, say loops, each such loop being a curve not cutting 
itself; and the case is thus reducible to that before considered, where the curve 
does not cut itself. Thus, to fix the ideas, let the curve be a figure of eight, the 
initial position of P being at the crossing, and let neither of the loops contain 
within it a point V. Then, as P passes continuously along one of the loops, re 
turning to its original position, each of the corresponding points P' describes a closed 
curve, which will be in the nature of a loop, viz. the initial and final directions of 
the motion of P not being continuous with each other, the initial and final directions 
of the motions of each point P' will not be continuous with each other, or there 
will be at the point P' an abrupt change in the direction of the curve. Similarly, 
as P describes the other loop of the figure of eight, each of the points P' will 
describe another loop; and the two loops belonging to the same point P' will unite 
together so as to form a figure of eight; viz. to the figure of eight described by P 
there will correspond figures of eight described by the n points P' respectively. 
9. But consider next the case where the two loops of the curve described by P 
include each of them one and the same point V. This implies that one of the two 
loops lies inside the other, or that the curve is what has been called an inloop 
curve. As P, which is in the first instance taken to be at the node, passes con 
tinuously along one of the loops and returns to its original position, there are two of 
the points P' such that the first of these passes from its original position to the 
original position of the second, and the second of them passes from its original 
position to the original position of the first of them. We have thus two arcs between 
these two points P'; but inasmuch as the initial and the final directions of motion of 
the point P are not continuous with each other, these two arcs are not continuous 
in direction at the two points P', but at each of these points P' the two arcs meet 
at an angle. As P describes the other loop, we have in like manner two arcs 
between the same two points P', these arcs at each of the points P' meeting at an