Plane Curves. 
457 
0 and 7 r only, we do not consider the negative values of a, as 
these would evidently give rise to negative values of t. There 
will be, then, \r/ 2] [s/2] values of a for which the correspond¬ 
ing t gives rise to one crossing, or to one double point. 
If we substitute the values of the positive a, which we shall 
from now on denote by a , in the equation for determining t', we 
find that t f will equal ir (hr + hs)/rs. For the sake of brevity we 
shall denote (hr + Its) by a'. 
From the symmetry of the values of the cosine, we might ex¬ 
pect that 7 r — air/rs would also give rise to a double point with 
the second crossing when t = ir — air/rs. Substituting these 
values in the equation of the curve we get 
/ 
a it\ 
x' = cos r \ 
/ 
a'Tt ' 
x = cos r 
Tt — 
- ) 5 
( 71 — 
\ 
rs / 
\ 
rs > 
= ± cos 
rait 
— ± cos 
ra'it 
rs 
f 
ait\ 
rs 
f 
a'rt' 
y' = cos S ( 
y = cos s 
( 7C ~ 
-) > 
Tt — 
rs / 
\ 
rs , 
sait 
sa'Tt 
= ± cos 
= ± cos 
rs rs 
which, as is easily seen from the form of a and a’, are the co¬ 
ordinates of double points. 
We have now to determine whether any of the double points 
given by 7 r — air/rs are the same as those given by air/rs. Three 
cases can arise: r and s can both be odd; r even and s odd; or 
s even and r odd. 
In the first case it is evident, from the form of the equations 
in (3), that the coordinates of the double points for 7 r — air/rs 
are all the negatives of those for air/rs, since cos [(2n1) ir — 0] 
= — cos 0. If, then, no two values of air/rs give rise to double 
points, each of which has as coordinates the negatives of the 
coordinates of the other, and the point £c = 0 , y — 0 , is not a 
double point given by air/rs, the double points of ir — air/rs are 
all distinct from those of air/rs. To show that no double point 
given by air/rs has as either coordinate, the negative of the cor¬ 
responding coordinate of another double point, given by another 
value of air/rs , we suppose that hr — hs gives rise to one value 
of a, and h'r — h's gives rise to another value of a. In order 
