can conceive in a general way; but the mathematical 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'; and the several notes which, according to his calculation, the same ring ought to give, were not in agreement with experiment. Indeed, researches of this kind, as conducted by Euler, and other authors, rather were, and were intended for, examples of analytical skill, than explanations of physical facts. James Bernoulli, after the publication of Chladni's experiments in 1787, attempted to solve the problem for plates, by treating 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 now before us as a prize-question*:-" To give the mathematical theory of the vibrations of elastic surfaces, and to compare it with experiment." Only one memoir was sent in as candidate for the prize; 2 Ib. vi. 596. 1 Fischer, vi. 587. 3 See Chladni, p. 474. HISTORY OF ACOUSTICS. and this was not crowned, though honourable mention was made of it. The formulæ of James Bernoulli were, according to M. Poisson's statement, defective, in consequence of his not taking into account the normal force which acts at the exterior boundary of the plate. The author of the anonymous memoir corrected this error, and calculated the note corresponding to various figures of the nodal lines; and he found an agreement with experiment sufficient to justify his theory. He had not, however, proved his fundamental equation, which M. Poisson demonstrated in a Memoir, read in 1814'. At a more recent period also, MM. Poisson and Cauchy (as well as a lady, Mlle. Sophie Germain,) have applied to this problem the artifices of the most improved analysis. M. Poisson determined the relation of the notes given by the longitudinal and the transverse vibrations of a rod; and solved the problem of vibrating circular plates when the nodal lines are concentric circles. In both these cases, the numerical agreement of his results with experience, seemed to confirm the justice of his fundamental views9. He proceeds upon the hypothesis, that elastic bodies are composed of separate particles held together by the attractive forces which they exert upon each other, and distended by the repulsive force of heat. M. Cauchy has also calculated the 10 6 Ib. 5 Poisson's Mém. in Ac. Sc. 1812, p. 169. • An. Chim. tom. xxxvi. 1827, p. 90. p. 220. transverse, longitudinal, and rotatory vibrations of elastic rods, and has obtained results agreeing closely with experiment through a considerable list of comparisons. The combined authority of two profound analysts, as MM. Poisson and Cauchy are, leads us to believe that, for the simpler cases of the vibrations of elastic bodies, mathematics has executed her task; but most of the more complex cases remain as yet unsubdued. 12 The two brothers, Ernest and William Weber, made many curious observations on undulations, which are contained in their "Wellenlehre,” (Doctrine of Waves,) published at Leipsig in 1825. They were led to suppose, (as Young had suggested at an earlier period,) that Chladni's figures of nodal lines in plates were to be accounted for by the superposition of undulations". Mr. Wheatstone has undertaken to account for Chladni's figures of vibrating square plates by this superposition of two or more simple and obviously allowable modes of nodal division, which have the same time of vibration. He assumes, for this purpose, certain "primary figures," containing only parallel nodal lines; and by combining these, first in twos, and then in fours, he obtains most of Chladni's observed figures, and accounts for their transitions and deviations from regularity. The principle of the superposition of vibrations is so solidly established as a mechanical truth, that we may consider an acoustical problem as satisfactorily disposed of, when it is reduced to that principle, as well as when it is solved by analytical mechanics : but at the same time we may recollect, that the right application and limitation of this law involves no small difficulty; and in this case, as in all advances in physical science, we cannot but wish to have the new ground which has been gained, gone over by some other person in some other manner; and thus secured to us as a permanent possession. Savart's Laws.-In what has preceded, the vibrations of bodies have been referred to certain general classes, the separation of which was suggested by observation; for example, the transverse, longitudinal, and rotatory, vibrations of rods. The transverse vibrations, in which the rod goes backwards and forwards across the line of its length, were the only ones noticed by the earlier acousticians: the others were principally brought into notice by Chladni. As we have already seen in the preceding pages, this classification serves to express important laws; as for instance, a law obtained by M. Poisson which gives the relation of the notes produced by the transverse and longitudinal vibrations of a rod. But this distinction was employed by M. Felix Savart to express laws of a more general kind; and then, as often happens in the progress of science, by pursuing these laws to a higher point of generality, the dis 13 Vibrations tournantes. tinction again seemed to vanish. A very few words will explain these steps. It was long ago known that vibrations may be communicated by contact. The distinction of transverse and longitudinal vibrations being established, Savart found that if one rod touch another perpendicularly, the longitudinal vibrations of the first occasion transverse vibrations in the second, and vice versa. This is the more remarkable, since the two sets of vibrations are not equal in rapidity, and therefore cannot sympathise in any obvious manner1. Savart found himself able to generalise this proposition, and to assert that in any combination of rods, strings, and lamina, at right angles to each other, the longitudinal and transverse vibrations affect respectively the rods in the one and other direction', so that when the horizontal rods, for example, vibrate in the one way, the vertical rods vibrate in the other. This law was thus expressed in terms of that classification of vibrations of which we have spoken. Yet we easily see that we may express it in a more general manner, without referring to that classification, by saying, that vibrations are communicated so as always to be parallel to their original direction. And by following it out in this shape by means of experiment, M. Savart was led, a short time afterwards, to deny that there is any essential distinction in these different kinds of vibration. "We are |