ON HARMONICS. 805 



jthe joint effecj>< doubled, or perhaps quadrupled, since it appears that the 

 'sfeno^r-dfsound ought to be estimated from the squares of the velocities 

 of the particles : but when the particular motions of the two sounds coun- 

 teract each other, both their effects are wholly destroyed. These combina- 

 tions 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 combina- 

 tion. We 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. 352.) 



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 quadruple force, and the sound gradually increases and diminishes in 

 continued succession 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 pro- 

 portion is known ; since the beats are usually slow enough to lqe reckoned, 

 although the vibrations themselves can never be distinguished. Thus, if one 

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

 acuter sound a single beat ^n?" every second, it is obvious that the second 

 sound must consist of 101 vibrations in a second. Again, if we have two 

 portions of a similar cord equally stretched, or two simple pipes, of which 

 the lengths are in the proportion of 15 to 16, they will produce a beat in 15 

 vibrations of the longer ;* and if we count the number of beats in 15 seconds, 

 we shall find the number of vibrations in a single second. The easiest way 

 of procuring two such strings or pipes, in practice, is to tune them by a 

 third, so that they may be respectively -t and -| O f its length ; the vibrations 

 of the third pipe in a second will also be equal to the number of beats of the 

 first two in 12 seconds. (Plate XXV. Fig. 353.) 



When the beats of two sounds are too frequent to be heard as distinct 



* For the times of performing a vibration are as the lengths of the cords or pipes, 

 and therefore 15 of the latter correspond to 16 of the former. Now an interval 

 between two beats is that interval which occurs between one relative state of the two 

 cords or pipes and the return to the same state. Hence this interval is that due to 

 16 vibrations of the shorter, or 15 of the longer. 



