390 LECTURE XXXIII. 



only. Thus, when two equal undulations, of equal frequency, coincide in 

 this manner, and when the particular motions are directed the same way 

 at the same time, the velocities in each direction are added together, and 

 the joint effect is doubled, or perhaps quadrupled, since it appears that the 

 strength of sound ought to be estimated from the squares of the veloci- 

 ties of the particles: but when the particular motions of the two sounds 

 counteract each other, both their effects are wholly destroyed. These com- 

 binations resemble the effects of the waves of water in similar circumstances, 

 which we have already examined, and they may be illustrated by drawing 

 two curved lines representing the motions which constitute the sounds, in 

 the same manner as we have already supposed them to be described, by a 

 vibrating particle, on a surface moving uniformly in a transverse direction ; 

 these figures being placed side by side, the joint effect may be represented 

 by a third curve drawn in such a direction as to be always in the middle 

 between the corresponding points of the first two. A similar result, but 

 still more strongly marked, may be obtained mechanically, by cutting two 

 boards or plates of any kind into the form of the curves, and then dividing 

 one of them into a number of thin pieces or sliders, by lines perpendicular 

 to the general direction of the curve, or to the termination of the plate 

 which is parallel to it: the bottom of these sliders being then placed on the 

 other curve, their general outline will represent the effect of the combination. 

 OWe may assume for this purpose the form of the harmonic curve , which 

 represents the motions of a body vibrating like a pendulum, and which 

 probably agrees very nearly with the purest and simplest sounds. (Plate 

 XXV. Fig. 359,.) 



If the two undulations differ a little from each other in frequency, they 

 alternately tend to destroy each other, and to acquire a double or perhaps a quadru- 

 ple force, and the sound gradually increases and diminishes in continued suc- 

 cession at equal intervals. This intension and remission is called a beat, and 

 furnishes us with a very accurate mode of determining the proportional 

 frequency of the vibrations, when the absolute frequency of one of them is 

 known, or the absolute frequency of both, when their proportion is known; 

 since the beats are usually slow enough to be reckoned, although the vibra- 

 tions themselves can never be distinguished. Thus, if one sound consisted 

 of 100 vibrations in a second, and produced with another acuter sound a single 



