Comstock—Real Singularities of Harmonic Curves. 453 
cal standpoint. He finds the algebraic equation to be of the 
2rth order (r being by supposition greater than s). He then 
determines the algebraic equation of the general curve. As the 
equation can be expressed by means of a parameter, its de¬ 
ficiency is zero and therefore it has the maximum number of 
double points, which is (2 r — l)(r — 1). These he next lo¬ 
cates. He then finds the class of the curve to be 2 (r — s), 
which gives for the number of double tangents 2 (r 4- s) 2 
— 3 (r + s) 4- 1. He then makes use of PHicker’s formulae and 
finds the number of inflexions, and finally discusses the curve 
on a Riemann’s surface. 
For the curve whose phase difference is zero, he finds the 
number of real double points to be 
(A) 
(r- 1) (s - 1) 
2 
and the number of real inflexions 
(B) r-s-1 
(r being greater than s). 
In the work which follows I shall obtain the number of real 
double points of the curve x = cos rt, y = cos st by determin¬ 
ing the number of pairs values of t which make the two val¬ 
ues of cc, corresponding to these values of t, equal and also makes 
the values of y , corresponding to the £’s, equal. The real in¬ 
flexions will be found by the consideration of the conditions which 
can cause the second derivative to change sign between differ¬ 
ent values of t. 
The object of the present paper is to determine the number 
of double points in the curve which results from compounding 
three harmonic motions, in phase, at right angles to each other. 
The equations to such a curve are 
x — cos nt y — cos rt z — cos st. 
The method used for these curves of double curvature is simi¬ 
lar to the method used in the case of the plane harmonic curves. 
