The mechanism of the ossicles of the ear and membrana tympani / by H. Helmholtz ; translated from the German, with the author's permission, by Albert H. Buck and Normand Smith.

  • Helmholtz, Hermann von, 1821-1894.
Date:
1873
Catalogue details

Licence: Public Domain Mark

Credit: The mechanism of the ossicles of the ear and membrana tympani / by H. Helmholtz ; translated from the German, with the author's permission, by Albert H. Buck and Normand Smith. Source: Wellcome Collection.

Provider: This material has been provided by The Royal College of Surgeons of England. The original may be consulted at The Royal College of Surgeons of England.

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    small compared with the others, as is the case, for instance, in disks, membranes, rods, and strings. The proof of these laws is easily deduced from the well-known laws respecting the form and mode of vibration of plane waves —so long, of course, as there is only question of plane waves of simple tones in masses of infinite extent. On the other hand, the influence of boundary-planes (Grenzfliichen) and of the last- named conditions has been elucidated by Kirchoff, in his trea- tise on the equilibrium and vibration of an infinitely thin elas- tic rod.* In this treatise, it is true, only the equilibrium of such elastie masses is taken into consideration, and it is there proven that forces, which are infinitely small compared with the constant of elasticity of the body, and which are brought to bear partly on the central and partly on the superficial por- tion of the elastic mass, cause only infinitely small relative dis- placements of such particles as lie within finite distance of each other, so that the differential quotients of the displacements, as taken from the co ordinates, also remain finite. On this last point the question chiefly hinges. For, if these differential quotients are finite magnitudes, then, in masses of infinitely small linear dimensions, the relative displacements of the indi- vidual particles are infinitely small, compared with the total absolute displacements which such masses experience. The above-mentioned law, which Kirchoff has demonstrated for the condition of equilibrium, the forces involved being infi- nitely small, may also, by means of d'Alembert's rule, be ap- plied to the condition of motion, provided the accelerations which the particles of the mass experience during motion are considered as the forces which disturb the elastic body. These now, when they belong to vibrations whose amplitude, com- pared with the length of the wave, is infinitely small, are them- selves infinitely small, and answer, therefore, to Kirchoff's ac- ceptation of infinitely small f disturbing forces. * Borchardt's Journal fur reine und angewandte Mathematik LVI., in § 1 of the treatise in question. t If A be the amplitude of vibration, n the number of vibrations of a sim- ple tone, t the time, and c a constant determining the phasis, then has s, the variable departure from the position of equilibrium, the following value : S — A sin \ 2 nnt + o \