Curves of Double Curvature. 
461 
In the third case if t f gives rise to the point xy\ 7 r — t' will 
give rise to — x\ y\ and the curve will be symmetrical with re* 
spect to the x axis. 
CURVES OF DOUBLE CURVATURE. 
11 . Periodicity of the curve 
x =* cos nt y = cos rt z = cos st. 
It is easy to see that the period of the curve of double curva¬ 
ture is 7 r, the same as in the case of a plane curve. It is also 
easily shown that if n , r , and s have a common factor 3 , the 
curve is a S-fold tracing. 
I 2. To find the number of double points in the curve 
x = cos nt y — cos rt z == cos st. 
Let us assume that n and r have a highest common factor c, 
n and s a highest common factor 8 , and r and s a highest com¬ 
mon factor y. 
If a double point arises, suppose the first passage of the 
curve through the point occurs when t — an/b. Then 
x — cos nan/b, y — cos ran/b , z == cos san/b 
give the coordinates of the double point. 
For the second passage of the curve through the point, t 
must be such that the values of x, y and z will be the same for 
the new t as for aWb. Call the new t , t'. Then t' must be of 
a form such that 
nt' = 2 hit ± nait/b 
rt' — 2k7C ± rait/b 
st =2jrt ± sait/b 
since for these values only can the two corresponding sets of 
values for tc, y and z be equal. 
The number t ' will then equal 
(1) 2 hit/n ± ait/b — 2kjc/r ± an/b = 2jn/s ± an/b 
