300 LECTURE XXXII. 



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

 of more difficult investigation than the vibrations of cords. A rfc4 nifty 

 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 extended by Euler,t some of whose conclusions have been cor- 

 rected by Riccati ; J and Chladni has compared them all with experinent. 

 The sounds produced by the same rod, either under different circum- 

 stances, 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 frequency of the vibrations of the 

 subordinate notes is expressed by the series of the squares of the 

 natural numbers, as 1, 4, 9, and 1G. 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 

 vibrations of rods of different kinds as in those of cords ; it can only 

 happen in particular cases that the rod will return after a complete 

 vibration to its original state, and these cases are probably such as seldom 

 occur in nature. 



The vibrations of plates differ from those of rods in the same manner as 

 the vibrations of membranes differ from those of cords, the vibrations 

 which cause the plate to bend in different directions being combined with 

 each other, and sometimes occasioning singular modifications. These vi- 

 brations may be traced through wonderful varieties by Professor Chladni' s 

 method of strewing dry sand on the plates, which, when they are caused to 

 vibrate by the operation of a bow, is collected into such lines as indicate 

 those parts, which remain either perfectly or very nearly at rest during the 

 vibrations. Dr. Hooke|| had employed a similar method, for showing the 

 nature of the vibrations of a bell, and it has sometimes been usual, in mili- 

 tary mining, to strew sand on a drum, and to judge, by the form in which 

 it arranges itself, of the quarter from which the tremors produced by 

 countermining proceed. (Plate XXV. Fig. 346... 348.) 



The vibrations of rings and of vessels are nearly connected with those of 

 plates, but they are modified in a manner which has not yet been suf- 



* Comm. Petr. iii. 62. Nov. Comm. Petr. xv. xvi. 257. 



t Comm. Petr. vii. 99. Nov. Comm. Petr. x. 243 ; xvii. 381 ; 1780, iv. 11.99. 

 Acta Petr. iii. I. 103. 



J Mem. della Soc. Jtal. i. 444. 



Entdeckungen iiber die Theorie des Klanges, Leipz. 1787. Acta Ac. Electr 

 Mogunt. Erford, 1796. Neue Schriften der Berl. Gesell. 1799. Traite d'Acous- 

 tique, 1809, PI. 3.. .7. Neue Beytrage zur Akustik, 1817. 



1| Birch's Hist, of the Roy. Soc. ii. 475. 



