322 ON THE GEOMETRICAL REPRESENTATION OF IMAGINARY VARIABLES [689 
viz. we have P/, P 2 describing the two portions of a bifid curve. If there were only 
a single branch-point V 12 , and therefore only a single barrier 0V 12 , then we might 
have through P a closed curve cutting 0V 12 once only, and including within it the 
point 0, but not including within it the point V 12 ; and here there ought not to be 
a bifid curve, but the points P/, P 2 ought to describe each of them a single curve. 
But suppose there are two points V 12 , and consequently two barriers 0F 12 (meeting 
in 0); then the closed curve, meeting once only a barrier 12, (viz. it meets only one 
such barrier, and that once only), must include within one and only one of the two 
points V 12 ; and in this case there ought to be a bifid curve. It is by such reasoning 
as this that I infer the foregoing theorem (No. 12), that the number of each set of 
points V 12 is even. 
16. We may consider how the suffixes are affected when, instead of the original 
system of barriers, we have a new system of barriers. I suppose that we have in 
the two cases respectively the same point Q, and the same suffixes for the points 
Qi> Qi, • • , Qn which correspond thereto. In the first case, passing from Q to an 
indefinitely near point 0, say the red 0, we draw from this point to the several 
points V a set of barriers, say the red barriers; while in the second case, passing 
from Q to an indefinitely near point 0, say the blue 0, we draw from this point 
to the several points V a set of barriers, say the blue barriers; and we then proceed 
as before, viz. in the first case, drawing from Q to the point S a curve which does 
not meet any of the red barriers, we determine accordingly the suffixes (say the red 
suffixes) of the several corresponding points S'; and in the second case, drawing in 
like manner from Q to S a curve which does not meet any of the blue barriers, 
we determine accordingly the suffixes (say the blue suffixes) of the same points S'. 
Now the curve drawn from Q to S so as not to cut any of the red barriers, and 
which is used for the determination of the red suffixes of the several points S', will 
in general cut certain of the blue barriers; and, by examining the suffixes of the 
blue barriers which are thus cut, we determine the blue suffixes of the same points 
S'; the result of course depending only on the situation of S in one or other of 
the regions formed by the red barriers and the blue barriers conjointly. In particular, 
the point S may be so situate that we can from Q to S draw a curve not meeting 
any red barrier or any blue barrier; and in this case the red suffixes and the blue 
suffixes are identical. 
17. We may imagine the first plane as consisting of n superimposed planes or 
sheets, say the sheets 1, 2Each barrier 12 is considered as a line drawn in 
the two sheets 1 and 2; and so on in other cases. The point P is considered as a 
set of superimposed points P 1} P 2 ,..,P n moving in the several sheets respectively; under 
the convention that P x moving in the sheet 1, and coming to cross a barrier 12, passes 
into the sheet 2 and becomes P 2 ; and the like in other cases. And this being so, we 
say that to a point P, considered as a point P a in the sheet a, there corresponds in the 
second plane one and only one point P a '; and that P moving continuously in any 
manner (subject to the change of sheet as just explained), each of the n corresponding 
points P' will also move continuously, and so that each such point P a ' will return