THE REAL SINGULARITIES OF HARMONIC CURVES 
OF THREE FREQUENCIES. 
ELTING H. COMSTOCK. 
INTRODUCTION. 
In the year 1800, Thomas Young 1 called attention to the paths 
traversed by any particle in a vibrating string. This led 
Wheatstone 2 in 1827 to study the vibrations of a rod fixed at 
one end. He found that if the rod was square or circular in 
section, the vibration of any point of the rod was in a plane 
passing through the axis of the rod, unless forced out by an ex¬ 
ternal disturbance. When the rod was rectangular in section 
the path described by any point was no longer a straight line, 
or more properly the arc of a circle, but the point followed a 
complex path, depending for its complexity, upon the ratio of the 
lengths of the two sides of the rectangle forming the section. 
Lissajous, 3 in 1857, in his well known memoir on “L’etude Op- 
tique des Mouvements Yibratoires, ” made an exhaustive study 
of these curves and from that fact they are now commonly desig¬ 
nated as “Lissa'ous’ Curves The equations of the Lissajous’ 
Curves are 
x = cos (rt + e x ) y — cos (st -f- e 3 ), 
t being the parameter, r/s the ratio of the number of vibra¬ 
tions parallel to the x axis to the number of those parallel to 
the y axis, and e 1 and e 2 the differences in phase. 
Wilhelm Braun 4 has studied these curves from the geometri- 
1 Phil. Trans, for 1800, pp. 106-150. 
9 Quart Jour. Sci. for 1827, Vol. I, pp. 344-351; also Poggend. Annal. for 
1827, Vol. X, pp. 470-480. 
3 Annales de Chimie et de Physique, 1857. 3© serie, tome LI. 
4 Dissertation Erlangen. Mathematische Annalen, 1875. Band VIII, s. 
567-573. 
