Plane Curves, 
459 
give rise to the same double point. Therefore, there are, when 
r is even and s odd, 
2 [r/2] [s/2] - [s/2] or 
double points in the curve. 
In case r is odd and s even it is easily seen that a similar 
reasoning holds, giving 
2 [r/2] [s/2] - [r/2] or 
double points in the curve. 
Thus we see that in any case which may arise the number of 
double points is equal to 
(r—l)(s—1) 
2 
The examination above made has shown that the double points 
which are counted twice are either all on the x axis (the case 
in which r is even), or are all on the y axis (the case in which 
s is even). In case r and s are both odd, double points can ex¬ 
ist on neither axis. In no case can double points exist simul¬ 
taneously on both axes. 
I 3. To determine the number of points of inflexion in the curve 
x = cos rt , y — cos st. — 
For convenience suppose that s is less than r. 
When t = mr/r 
.... d 2 x _ r 2 & cos mt 
' dy 2 —s 3 sin 2 nsit/r 
When t — (n + 1)? r/r 
d 2 x _ r 8 scos ( n -j- 1)tc 
' dy 3 —s 3 sin 3 (n-j-1) sit/r 
Since the sine appearing in the denominator of the second 
derivative is always squared, its value will not change sign for 
any value of n, and the sign of the second derivative will then 
depend only on that of the cosine in the numerator. Now 
cos mr = — cos (n + 1 )tt, so that the sign of the second deriva¬ 
tive in (1) is the negative of that in (2), therefore the curva¬ 
ture changes between t — mr/r and ( n + 1 ) 7 r/r. It is obvious 
