320 ON THE- GEOMETRICAL REPRESENTATION OF IMAGINARY VARIABLES [689
angle ; but they join on to the first-mentioned two arcs in such manner as to form
two ovals intersecting each other in the two points P'. Corresponding to the inloop
curve described by P, we have this pair of intersecting ovals described by two of
the points P', and n — 2 other curves described by the other points P', and being
each of them (I assume) an inloop curve.
10. If we attend only to one of the two intersecting ovals, we have in the
first plane an inloop curve, and corresponding thereto in the second plane an oval
passing through two of the points P' which correspond to the node P of the inloop
curve. Interchanging the two planes, and writing Q instead of P, we have in the
first plane an oval passing through two of the points Q which correspond to a point
Q'; and corresponding to this oval we have in the second plane an inloop curve
having this point Q for its node, viz. these are the corresponding figures mentioned
in No. 3.
11. Consider a given point Q; and let the corresponding points Q' be called
(selecting the suffixes at pleasure) Qi, Qi,.., Q n '. Taking then a point 0 indefinitely
near to Q, the corresponding points O' will be indefinitely near to Qi, Qi, .., Q n '
respectively, and they will be called Oi, 0,', .., 0 n ' accordingly. It is to be observed
that by the indefinitely near point 0 is meant a point such that the distance from
0 to Q is indefinitely small in comparison with the distance of either of these points
from any point V ; so that we cannot have from Q to 0 two indefinitely short paths
including between them a point V ; or say so that the indefinitely short path from
Q to 0 is determinate.
Proceeding in this manner from Q to 0, and so through a succession of indefinitely
near points to a distant point S, we seem to determine the suffixes of the corre
sponding points S'; but, by what precedes, it appears that such determination for a
point S is dependent on the path from Q to S; and consequently that we do not
thus obtain a proper determination of the suffixes of the points S'. In fact, if we
were to pass from Q by a path including one or more of the points V back to Q,
we should obtain for the several points Q' respectively suffixes which are in general
different from the suffixes originally given to these points respectively.
12. The difficulty is got over as follows:—Considering as before the given point
Q, and calling the corresponding points Qi, Qi, .., Q n ' at pleasure, we pass from Q to
the indefinitely near point 0, and thence, by so many paths chosen at pleasure, to
the several branch-points V ; these paths from 0 to the several points V are called
barriers. To fix the ideas, we may consider these as non-autotomic non-intersecting
lines drawn from 0 to the several points V. Consider the barrier from 0 to one of
these points F ; as P passes along this barrier from 0 to V, two of the corre
sponding points P' will pass from two of the corresponding points 0' to the corre
sponding cross-point (W); the paths of these two points are called the counter-bamer
corresponding to the barrier in question ; and we have thus in the second plane a
system of counter-barriers, each drawn from two points 0' to meet in a point ( W').
By what precedes, the points 0' have each of them a determinate suffix ; a counter
barrier is thus drawn from two points with given suffixes, suppose Oi and Oi, to a
