292 PROCEEDINGS OP THE AMERICAN ACADEMY 



XV. 



ANTICIPATION OF THE LISSAJOUS CURVES. 

 By Joseph Lovering, 



Hollis Professor in Harvard College. 

 Presented January 12, 1881. 



In 1857, Lissajous communicated to the French Academy of Sciences 

 his " Memoire sur I'Etude Optique des Mouvements Vibratoires." * By 

 attaching a mirror to each of two tuning-forks, and reflecting a pencil 

 of light successively from the two mirrors, the resultant motion of the 

 forks was exhibited on a greatly amplified scale. When the vibrations 

 of the tuning-forks were in parallel planes, and the musical interval 

 between them was not exactly in tune, the consequent beats were 

 manifested to the eye, even ^fter they had become insensible to the 

 ear. When the two motions were in rectangular planes, the extremity 

 of the pencil of light described some one of a group of curves, rising 

 to higher orders as the ratio which expressed the musical interval be- 

 came less simple. Each of these groups could be easily resolved into 

 eight distinct varieties, each variety depending on a definite difference 

 of phase in the two movements. Inasmuch, however, as there are 

 innumerable differences of phase between any two well-marked limits, 

 there must be an endless series of curves in each group, through 

 which each variety flows gradually into the next. Lissajous studied 

 the subject analytically as well as experimentally, and then applied 

 it to his optical method of tuning notes, especially those of tuning 

 forks, to any required interval. If the vibrations of the forks were 

 true to the precise ratio of this interval, the resultant orbit of the twice- 

 reflected ray would be invariable : and it would be one or another 

 variety in the corresponding group of curves, as the original difference 

 of phase was smaller or larger. If the movements of the two forks 

 did not conform to the exact ratio, the original difference of phase 

 would change with each successive vibration, and the wliole group of 



* Annales ile Chim. et iJo I'lij's., vol. li. 18J7. 



