Booh I 
EVEN AND ODD CIRCUITS 
55 
Chap. IV 
an even circuit 
/ its extremities, 
M, only one of 
w arc. 
bhe second sort 
iommon point, 
another on the 
that shared no 
meeting A 1 ,B 1 
necting A 2 ,B 2 , 
deli is contrary 
'egard to an even 
gr on the circuit, 
ts follow in the 
at most one arc 
2 intersections 
no arc of one is 
3 shall see later 
the successive 
ssive pair. Pick 
which contains 
ae with one arc 
scond sort with 
d by 8]. If the 
if the first will 
nd by 6]. If the 
) of the second 
e to consider is 
)e even we may 
I and we choose 
action, and that 
lions, then this 
ircuit composed 
in absurdity, 
e present an arc 
he h successive 
intersections will determine Jc successive pairs of arcs containing 
no other intersections, each pair dividing the plane. Let the 
equations of the curves on which the circuits lie be 
/1 = 0 A=0. 
Let us write f =fj 2 + e( f> = °> 
where e is infinitesimal. The curve / is infinitely near the 
original curves and meets them only where they meet f or at 
infinity. If we take for f a curve which meets neither circuit 
in a real point, and e so that f contains a point inside one of the 
even circuits, it will contain a point inside each of the k just 
described and have, in fact, k even circuits infinitely near the 
others. These circuits, shall be said to be obtained from the 
others by the method of ‘small variations’.* 
Suppose that we have two circuits which lie infinitely near 
to one another, i.e. they are corresponding parts of two curves 
whose equations differ infinitesimally. A curve meeting one in 
real points will meet the other in the same number of them. 
Their intersections, if they have any, must follow in the same 
order on the two curves, and both must be even or odd 
together. Moreover, one could not make an arc of the second 
sort with regard to the other, for an odd circuit lying outside of 
one, and hence, outside of the other, would fail to meet both, 
which is not possible if one have an arc of the second sort with 
regard to the other by 5]. 
Theorem 9] If two infinitely near circuits intersect in a number 
of points, these will follow in the same order on the two, and we 
may obtain k even circuits from them by the method of small 
variations. 
Let us see what is the maximum number of circuits obtained 
by this method. We shall proceed to prove 
Theorem 10] If two curves f x ,f 2 have N simple circuits, and k 
intersections, the maximum number of circuits obtained from them 
by the method of small variations is N A-k—2, and this number is 
only attained when all the intersections lie on one pair of circuits 
arranged in the same order. 
If there be s circuits each intersecting none on the other curve, 
these will be replaced by an equal number of like circuits. 
* Of. Brusotti 1 for this definition.