84 M. J. Antoine oji the Multiple Sounds 



anniilletl, and where they will become null again successively 

 for each point, if the decrease in amplitude of the vibrations 

 is neglected. 



From these considerations it follows, that during the com- 

 plete vibration of the string, four distinct loci are formed in 

 which the points of the string arrive successively without ve- 

 locity, after having traversed the intermediate positions with a 

 greater or less rapidity. Thence results the appearance of 

 four distinct strings, curved as the lines of the figure indicate, 

 and consequently the explanation of the optical pbeenomenon 

 which is observed. It follows from these same considerations, 

 that during an entire oscillation, the string strikes the air four 

 different times, in passing from AaCZ'B to AfC/'B, from this 

 last position to A^H^B, then to the position AwH?iB, and 

 lastly, to the first position AaC^B. These four shocks are 

 not identical : in the first and the third, the two halves of the 

 string strike the air in a contrary direction, whilst in the second 

 and the fourth, the string strikes the air in the same direction at 

 all the points, nearly as if it moved without being divided. 

 Thence it results that in the series of shocks which the tym- 

 panum of the ear receives, at each second shock the impres- 

 sion made on us must have something distinctive, and the 

 periodical return of this phcenomenon brings the simultaneous 

 sensation of two sounds, the one the octave to the other. This 

 is precisely the fact which observation indicates and which 

 had to be explained. 



All the preceding observations may be confirmed by com- 

 pleting in the following manner the experiment which we 

 have just described. 



If, by a suitable management of the bow, the octave of the 

 fundamental tone is weakened, the spindles AaC^*, CbBf, 

 A^H;«, HArBw will diminish in width, and will end in being 

 imperceptible, when the octave has become very weak. 



If, on the contrary, you weaken the fundamental sound, 

 the spindles will persist, but the parts C and H will approach, 

 will tend towards being confounded, and will in fact be so, 

 when the fundamental sound is silenced. 



The form of the curves will be easily determined upon 

 which the different points of the string arrive successively with- 

 out velocity, and the duration of the motion of each point in 

 passing from one curve to the other, by aid of the formula 



. vx . irat - . 2vx . Q,iTat 

 g=sa sin — sm — — 1-/3 sm -t— sin — ^, 



which represents one of the possible movements of the string, 

 and may specially represent that which we have just analysed. 



