The limited nature of the solution given by Dr Tay lor, induced D'Alembert to apply himself to the same question. The result of his labours was given in the Berlin Memoirs for the year 1750. In that volume, D'Alembert has, by the mode of partial differences, given a solution of this problem, which embraces all the initial forms of a chord, in which the law of continu ity takes place ; and has shown, that there is an infinite number of curves, different from that discovered by Dr Taylor, each endowed with this property, that all its points arrive simultaneously at its axis. Shortly after, Euler gave another solution, founded on similar prin ciples, which led him to a construction more general than that which D'Alembert had employed. It was ob jected to the generality of this construction, first by D'Alembert, and afterwards by La Grange, that the principles on which it is founded necessarily limit its application to those cases in which the initial form of a string is a continued curve. Euler, with a greatness of mind of which we have hut ft w examples, acknowledged the justice of the remarks which the latter of these ma thematicians had made against the generality of his con struct ion.
Daniel Bernoulli attempted to extend Dr Taylor's solution to all possible initial forms, by conceiving them to be either harmonic curves, or produced by a combina tion of several subordinate harmonic curves. This sup position enabled him to give a solution of the problem of vibrating strings, equally extensive in its application with those which can be legitimately deduced from the methods of either D'Alembert or Euler. These three mathematicians have equally filed in that their equations to all possible cases of a vibrating chord. To remedy this defect, La Grange im,estin ted this question by a mode perfectly new, and totally independent of the hypothesis, that the initial Oman of the vibrating chord is subjected to any law of cold Mu ity ; and therefore his conclusion~ must be considered as independent of any such law. 11e considers a vibrat ing chord under two views, either as composed of a finite or an infinite number of parth les. In the cornier case, analysis conducts him to a general theory, the same with that which we have mentioned tffiov c as invented by Daniel Bernoulli. In the latter case, his conclusions are exactly the same w ith those w hich Euler had drawn from sources not so legitimate.
Daniel Bernoulli, subsequent to the publication of his essay on vibrating chords, investigated the lateral vibra tions of an elastic rod fixed at one extremity; and de termined the vibrations of a column of air contained in a pipe. The conclusions at which lie has arrived, have, when brought to the test of experiment, been found ac curate, though deduced from suppositions which are not considered as perfectly just. Euler and La Grange
have also prosecuted this latter subject, by methods similar to those which they have employed in the pro blem of vibrating chords. The vibrations of several other bodies have been considered by both Bernoulli and Euler, and the results of the latter corrected in some instances by Riccati.
Sir Isaac Newton was the first who investigated, with any precision, the propagation of sound. Ilis reason ings on this subject were always considered as extreme ly difficult and obscure, and have been shown, first by AI. Cramer, and afterwards more fully by La Grange, to be in some respects faulty ; but by that good fortune which attended him in all his researches, his conclu sions are accurate, and have been confirmed by the in quiries of subsequent philosophers.
Several other mathematicians attempted this subject with no better success than had attended the efforts of sir Isaac Newton ; their methods being founded on such erroneous principles, or their calculations being so embarrassed with infinite series, as to remove all confidence in their conclusions.
At length, about the year 1759, both La Grange and Euler succeeded in giving solutions of this problem, unobjectionable in their principles, and extensive in their application; thus subjecting to analysis a problem in volved in such difficulties, as seemed to place it wholly above the reach of mathematical investigation.
One circumstance, however, rendered the labours of these philosophers not quite satisfactory. The velocity which theory uniformly attributes to sound, is found to differ considerably from experience. The cause of this difference La Place has lately suggested to lie an increase in the elasticity of the air, produced by the heat evolved during, the condensation to which it is sub jected in transmitting an undulation.
Biot has examined what increase of elasticity would be required to render theory consistent with observa tion ; and has found it to differ very little from what should take place agreeably to the experiments of Mr Dalton on air, removing by this result the chief dif ficulty in the theory of sound.
The science of Acoustics is indebted to several other philosophers, who have laboured in the experimental de partment of this science.
About the year 1651, Soland made the first speaking trumpet from the description given by Kircher, of the tube which Alexander was supposed to have used in commanding his armies. 'Moreland, however, by draw ing the attention of philosophers to that which he con structed in 1671, had the merit of being the first who made this instrument really known, and applied to use.