454 Comstock—Real Singularities of Harmonic Curves. 
If n and r have a common factor c, n and s a factor 8, and r 
and s a factor y, I find that the number of real double points is 
(C) 
(n — 1 ) ir — 1 ) , (r — 1 ) (8 — 1 ) , (s — 1 ) {e — 1 ) 
2 
+ 
+ 
(r “ 1) (8 -1) (8 -1) (fi - i) (8 - 1 ) (r -1) 
2 2 2 
If £ is equal to unity, that is, if n and r are prime, this num¬ 
ber will evidently reduce to 
(D) 
(n-l)(y-l) . (r~l)(8~l) (y - 1 ) (8 - 1 ) 
2 
+ 
2 
If £ is equal to unity and 8 is equal to unity, that is, if n is 
prime to both r and s, the number reduces to 
(E) 
(n — 1 ) (y — 1 ) 
2 
If all three numbers ?i, r, and « are prime to each other, 
there are no double points, for then c, S, and y will each equal 
unity and the formula above written reduces to zero. In each 
of the last three special cases the number of double points can 
be found by an independent method, without using the general 
formula. 
The conditions of symmetry for both classes of curves have 
been found and are given for all cases that can arise. The con¬ 
ditions for the plane curves were given in Lissajous’ original 
article, but I believe those for the curves of double curvature 
have never before been worked out. 
PLANE CURVES. 
1 1. Periodicity of the curve .— 
It is easy to show that the curve x = cos rt, y == cos st is 
completed in a cycle in which t passes from 0 to tt. 
It is also easily shown that if r = 8 a and s = 8 /?, (8 being 
the highest common factor of r and s) the curve x — cos rt, 
y = cos st is a 8 fold trace of the curve x = cos at t y = cos fit. 
