VIBRATION OF BODIES IN GENERAL. 359 



which divide solid bodies, and masses of air, when 

 in a state of vibration. 



The dependence of such vibrations upon their 

 physical cause, the elasticity of the substance, we 

 can conceive in a general way ; but the mathema- 

 tical theory of such cases, is, as might be supposed, 

 very difficult, even if we confine ourselves to the 

 obvious question of the mechanical possibility of 

 these different modes of vibration, and leave out of 

 consideration their dependence upon the mode of 

 excitation. The transverse vibrations of elastic 

 rods, plates, and rings, had been considered by Euler 

 in 1779 ; but his calculations concerning plates had 

 foretold only a small part of the curious phenomena 

 observed by Chladni 1 ; and the several notes which, 

 according to his calculation, the same ring ought 

 to give, were not in agreement with experiment 2 . 

 Indeed, researches of this kind, as conducted by 

 Euler, and other authors 3 , rather were, and were 

 intended for, examples of analytical skill, than ex- 

 planations of physical facts. James Bernoulli, after 

 the publication of Chladni's experiments in 1787, 

 attempted to solve the problem for plates, by treat- 

 ing a plate as a collection of fibres ; but, as Chladni 

 observes, the justice of this mode of conception is 

 disproved, by the disagreement of its results with 

 experiment. 



The Institute of France, which had approved of 

 Chladni's labours, proposed, in 1809, the problem 



1 Fischer, vi. 587. 2 Ib. vi. 596. ' See Chladni, p. 474. 



