384 



LECTURE XXXII. 



intermediate direction, or a revolution, in a circle or in an ellipsis. But 

 besides these compound vibrations of the whole chord, it is also frequently- 

 agitated by subordinate vibrations, which constitute harmonic notes of 

 different kinds, so that the whole effect becomes very intricate; as we 

 may observe by a microscopic inspection of any luminous point on the sur- 

 face of the chord, for instance the reflection of a candle in the coil of a fine 

 wire wound round it. The velocity of the motion is such that the path of 

 the luminous point is marked by a line of light, in the same manner as when 

 a burning coal is whirled round; and the figures, thus described, are not 

 only different at different parts of the same chordjbutthey often pass through 

 an amusing variety of forms during the progress of the vibration; they 

 also vary considerably according to the mode in which that vibration is 

 excited. (Plate XXV. Fig. 344, 345.) 



The vibrations immediately dependent on elasticity are those of rods, 

 plates, rings,and vessels. These admit of much greater variety, and are of more 

 difficult investigation than the vibrations of chords. A rod may be either 

 wholly loose, or fixed at one end only, or at both ; and it may either be 

 loosely fixed, in situation only, or firmly fixed, in direction as well as in 

 situation; and these conditions may be variously combined with each other; 

 the rod may also have a variety of secondary vibrations besides the principal 

 or fundamental sound. All these cases have been examined by various 

 mathematicians: the subject was begun by Daniel Bernoulli, and much ex- 

 tended by Euler, some of whose conclusions have been corrected by Riccati; 

 and Chladni has compared them all with experiment. The sounds produced 

 by the same rod, either under different circumstances, or as harmonics which 

 may be heard at the same time, are scarcely ever related to each other in any 

 simple proportion, except that when a rod is loosely fixed at both ends, the frequen- 

 cy of the vibrations of the subordinate notes is expressed by the series of the 

 squares of the natural numbers,as l,4,9,and l6. But the times occupied by any 

 similar vibrations of rods, similarly circumstanced, are always directly as 

 the squares of their lengths, and inversely as their depths. When the rod 

 is wholly at liberty, two at least of its points must be at rest, and these 

 are at the distance of about one fifth of its length from either end: in the 

 next sound of the same rod, the middle point is at rest, with two others near 

 the ends. There is by no means the same regularity in the progress of the 



