GIVE THE IMAGINARY ROOTS OF AN ALGEBRAIC EQUATION. 469 
of double points of Q lie between them. This explains why the rule does not 
apply, when the curves intersect as they do in figs. 4 and 5. 
Looking again at fig. 7 we see that no two branches of P and Q can inter- 
sect again when prolonged ; for, if they did, there would be two intersections 
of P and Q on one branch, either of P or Q, without any double point between. 
It is not at first sight obvious how far this conclusion would be modified by the 
existence of another branch of Q passing through the intersection, as the dotted 
curve in fig. 8. It is necessary therefore to consider multiple points of this 
kind, that is to say, points through which pass two or more branches of one 
curve and one or more of the other. First consider the case of two branches of 
P and two of Q passing through one point and arranged alternately, 
as shown in fig. 9. The simplest way in which such a point can arise is by two 
branches of P moving up towards a branch of Q, as if in fig. 10 the points A 
and B should move up and coincide with C. In that case two points of inter- 
section of P and Q would coincide, and the multiple point in fig. 9 is therefore 
to be reckoned as two intersections of P and Q. Therule gives us /=4, m=0, 
4 (J—m) =2, which is correct. If the branches are not arranged alternately, 
but occur as in fig. 11, the simplest way in which it can arise is by two branches 
of P joining to form a double point which is already a double point onQ. In this 
case it seems that the point should, notwithstanding its apparent complexity, 
not be counted as an intersection at all. This agrees with the rule, which gives 
/—2, m=2,4(J—m)=0. Similarly, if we have a point through which pass one 
branch of P and two branches of Q, this may be considered to have arisen from 
the union of two branches of Q which do not cut P, as the dotted curves in fig. 
12, and the point will therefore not count as an intersection ; this being again in 
agreement with the rule, which gives /=1, m=1, 4(/—m)=0. In this way of 
reckoning intersections, the rule applies to such a curve as is shown in fig. 8. 
Here, if we take a contour to include the two points A, B, but not C, the rule 
gives /=3, m=1, 4(/—m)=1; and this is correct, since, as we have seen, B 
does not count as an intersection. If the contour is drawn so as to include 
also C, we shall have /=4, m=0, 4 (/—m)=2, which also is correct. 
Resuming consideration of figs. 7 and 8, we see that it follows from what 
we have proved that no branch of P which intersects Q can re-enter upon itself 
and form a closed curve, but it must always have two infinite branches; and 
the same is true of Q. We next observe that the rule will not hold unless the 
arrangement of the infinite branches follows a law similar to that we have seen 
to prevail among the intersections. If in figs. 13 and 14 we draw a contour to 
include the points of intersection A and B, we shall have in fig. 13, /=2, m=2, 
4(/—m)=0; and in fig. 14, 7=4, m=0, 4 (7—m)=2; or the rule applies in the 
latter case, but not in the former. If, however, in fig. 13 we draw between 
A and B an infinite branch either of P or Q or an odd number of such branches 
