TRANSACTIONS OF THE SECTIONS. 5 



Its immediate object is to determine to what extent our present knowledge of the 

 condition and properties of elastic bodies enables us to use experiments on the velocity 

 of sound in them as data for calculating the elasticity of the materials. If it were 

 possible to ascertain the velocity of propagation of vibratory movements along the 

 axes of elasticity of an indefinitely extended mass of any substance, we could at once 

 calculate the coefficients of elasticity of the substance ; for in such a mass we can 

 assign the directions of molecular oscillation corresponding to each direction of propa- 

 gation, and consequently the nature of the elastic force called into play. Such experi- 

 ments, however, are possible in air and water only. For other substances, the best 

 data which it is practicable to obtain are experiments on the velocity of sound along 

 prismatic or cylindrical masses. 



The author in the first place integrates the equations of small molecular oscilla- 

 tions in elastic bodies of limited dimensions, and of any structure, and afterwards 

 investigates the special results to which they lead in the case of uncrystallized media 

 generally, and of prismatic bodies of liquid in particular. The principal positive con- 

 clusions arrived at may be summed up as follows : — 



I. In liquid and solid bodies of limited dimensions, the freedom of lateral motion 

 possessed by the particles causes vibrations to be propagated less rapidly than in an 

 unlimited mass. 



II. The symbolical expressions for vibrations in limited bodies are distinguished 

 by containing exponential functions of the coordinates as factors ; and the retardation 

 referred to depends on the coefficients of the coordinates in the exponents of those 

 functions, which coefficients depend on the molecular condition of the body's surface, 

 a condition as yet imperfectly understood. [Exponential functions in the equations 

 of small oscillations have not hitherto been used, except in the theory of waves pro- 

 pagated by gravitation, by Mr. Green, in that of total reflexion, and by Professor 

 Stokes, in investigating the possible eflfect of radiation on the velocity of sound.] 



In an uncrystallized body in particular, if A represent the coefficient of longitu- 

 dinal elasticity ; that is to say, the quantity by which an elongation or compression, 

 without lateral yielding, must be multiplied in order to give the pressure per unit of 

 area to which it corresponds*; if D denote the weight of unity of bulk of the sub- 

 stance, and g the accelerating force of gravity, then, while \/ -^ denotes the velo- 

 city of sound in an unlimited expanse of the substance, that in a body of limited 

 dimensions is denoted ^Y \/^ . (i—h''), where A is a quantity depending on the 



figure and molecular condition of the body's surface, and entering into the exponents 

 of the exponential factors in the equations of motion. The trajectory or orbit of an 

 oscillating particle is, generally speaking, a straight line in an unlimited expanse, 

 and an ellipse in a limited body. 



III. If we adopt the hypothetical principle, that at the free surface of a vibrating 

 mass of liquid the normal pressure depending on the mutual actions of atomic centres 

 only is always null, then we deduce from theory that the ratio VI — h^ : 1 of the 

 velocity of sound along a mass of fluid contained in a rectangular trough to that in 

 an unlimited expanse, is \^2 : \^3, that ratio being independent of the specific 

 rigidity of the liquid, provided only that it has some amount, however small. This 

 theoretical conclusion is exactly verified by a comparison of the numerous experiments 

 of M. Wertheim on the velocity of sound in water at various temperatures, from 15° 

 to 60° Centigrade, in solutions of various salts, in alcohol, turpentine and aether (Ann. 

 de Ch. et de Phys. Ser. III. tom. xxiii.), with those of M. Grassi on the compressi- 

 bility of the same substances (Comptes Rendus, xix. p. 153), and with the experi- 

 ments of MM. Colladen and Sturm on the velocity of sound in an expanse of water. 

 To these may be added the following negative conclusions : — 



IV. We are not warranted in concluding from M. Wertheim's experiments (as 

 he is disposed to do) that liquids possess a momentary rigidity as great as that of 



* The quantity A is greater than the modulus of elasticity, because, in computing the 

 latter quantity, the particles are supposed at hberty to yield laterally. 



