Plane Curves. 
455 
§ 2. To determine the number of double points in the curve x = 
cos rt , y = cos st , r and s being considered prime to each other. 
Since, when the numbers r and s are prime to each other, the 
curve is traversed but once in the interval from t = 0 to t = 7 r, 
in order that a double point may exist there must be two val¬ 
ues of t less than 7 r for which the corresponding values of x and 
y are equal. Suppose a double point occurs when t = an/b, in 
which a > b, and a and b are prime to each other. The co-ordi¬ 
nates of the double point will then be 
rare san 
x — cos . , y — cos ^ ' 
Of course b cannot be a factor of r or of s. If we call t the 
second value of t corresponding to the double point, we know 
from the properties of the cosine that t must be of such form 
that the co-ordinates of the double point will be given by 
= cos 
2hn ± 
r a 7t\ 
~b~) ’ 
Therefore € must equal both 
y 
2 h 
r 
/ cyl sa7t\ 
= COS ( 2JC7C ± -jp-J 
a\ n /2 k a\ 
± y)* and (— ± y)*. 
Since € can have but one value, these two values must be equal 
to each other, that is, 
m 2_h _2 _k a_ 
() ~ ± b — T ± b 
The signs before a/b on each side of this equation cannot be 
the same, for if they were the equation would become h/r = k/s. 
This, since r and s are prime to each other, can only be true 
when h and k are equal integral multiples of r and s respect¬ 
ively. Let h — er and k = es. Then t' equals 2e?r ± air/b which, 
since by hypothesis a is less than b, would be greater than tt; 
but t cannot be greater than tt for a single tracing of the 
curve. Hence h and k can never be equal integral multiples of 
r and s respectively, and the signs before a/b on each side of 
the equation ( 1 ) must differ. 
