56 Mr. S. Taylor on Variations of Pitch in Beats. 



velocity of the impinging molecule. This may be proved as 

 follows. 



By a well-known formula^ if Wj and W^ be the weights of two 

 imperfectly elastic impinging bodies, and Y^, Vg the velocities of 

 their impact, and e the measure of their elasticity, and u^ half 

 the vis viva lost by W^ in its impact, then 



,, _ (iH-^)w,w,(y,+y,) 



{2W,yi + (1 - e) W^Y, ± (l-f .) W2V2}. 



(See ^^ Mechanical Principles of Engineering and Architecture/^ 

 by the author of this paper, Art. 440.) 

 " If Y2=0, or if one of the bodies be at rest, 



-.= |(^^§.{3W, + (l-.)WJV?; 



u^ varies, therefore, in this case as Yj — that is, as the square of 

 the velocity of the impinging body. 



Therefore the work lost per unit of time by the impact of the 

 molecules of the moving film on those of the fixed one is pro- 

 portional to (KY)N2 or to KY^. Let it = XjKY^ ; 



.-. PKY = K^iY + \KY^ 

 and 



lY. On Variations of Pitch in Beats. By Sedley Taylor, Esq.j 



late Felloiu of Trinitij College^ Cambridge^. 



PROFESSOR HELMHOLTZ, as all readers of his great 

 work, Die Lehre von den Tonempfindungen^ are aware, has 

 proved that discord in music invariably arises from certain phe- 

 nomena called beats, which were well known to previous writers 

 on Acoustics, though their extreme importance had not been re- 

 cognized. Beats occur whenever two musical sounds of slightly 

 different pitch are simultaneously produced. Helmholtz describes 

 their effect in the following words : — ^'^ The intensity of the sound 

 becomes alternately strong and weak in regular succession/' 

 Up to the second edition of the Tonempfindungen no variation 

 save that of intensity was mentioned. In the third edition, how- 

 ever, the author states that a slight oscillation in pitch on the 

 part of the beating sound is likewise perceptible, adding that this 

 fact was communicated to him by M. Gueroultf. But, although 



* Communicated by the Author. 



t P. 259. References are to the third edition, Braunschweig, 18/0. 



