468 THOMAS BOND SPRAGUE ON THE CURVES WHOSE INTERSECTIONS 
assuming for the present that f(v)=0 has no equal roots, CAucuy’s theorem 
asserts that we can ascertain how many intersections of the curves lie within 
any assigned area, by observing how often p:gq passes through the value 0 and 
changes its sign, when we substitute for « and y the co-ordinates of a moving 
point that traverses the contour of the area in the positive direction. If p:¢ so 
changes sign 7 times from + to —, and m times from — to +, then the number 
of points of intersection is } (7—m). Calling the two curves P and Q, it is 
obvious that p:g passes through the value 0 and changes sign, only when the 
moving point crosses the curve P. When the moving point crosses the curve Q, 
p:q changes sign; but, as it passes through the value ow, this change of sign is 
not counted. Before proceeding further let us consider whether it is true of 
ail curves that the number of intersections is determined by the above formula, 
or for what particular kinds of curves it holds good. 
Suppose the point to move round the rectangle in any one of the figures (1), 
(2), (8) ; then in the first case it crosses the curve P twice, but on one occasion 
p:q changes sign from + to —, and on the other from — to +. There is 
therefore, according to the rule, no point of intersection of P and Q within the 
rectangle ; and this is the case. In fig. 2 the point also crosses P twice, and the 
same changes of sign occur, so that here again, according to the rule, there is 
no intersection within the rectangle; and this is the fact. In fig. 3 the point 
crosses P twice, and on each occasion there is a change of sign from + to —,so 
that /=2, m=0, and the rule gives us one point of intersection, which is correct. 
If, however, we suppose the curves P and Q to intersect in two points within 
the rectangle, as in figs. 4 and 5, we see that the rule does not hold good. In 
fiz. 4, the point crosses P twice, and we have /=1, m=1; in fig. 5, the poimt 
crosses P four times, and we have /=2, m=2; in both cases, therefore, the rule 
would give us no intersections, whereas there are two. This shows that the 
rule does not apply to all kinds of curves, and it is not difficult to determine 
the general characteristics of the curves to which it does apply. 
Suppose a branch of P to cut two branches of Q, thus giving two inter- 
sections, then, if we draw a rectangle to include these intersections, as in fig. 6, 
we see that the rule does not bold good. If, however, a second branch of P 
lies between the two branches of Q, as in fig. 7, the proper sequence of signs 
will be obtained, and the rule will hold good. In this case we get /=4, m=0, 
3 (J—m) =2, which is the actual number of intersections of P and Q. Proceed- 
ing in the same way we see that, if there are an odd number of branches 
of P between the two branches of Q, the rule holds good, but not if the number - 
of branches is even; and we conclude that, when the rule applies, if two inter- 
sections of P and Q occur on a branch of P, an odd number of double points 
of P lie between them. Similar reasoning shows that if two intersections of 
P and Q occur on a branch of Q, the rule will not apply unless an odd number 
