The Aether as an Elastic Solid. 139 



the (vector) displacement of the particle whose undisturbed 

 position is (x, y, z], and if p denote the density of the medium, 

 the equation of motion is 



p = - 3n grad div e - n curl curl e, 



ot 



where n denotes a constant which measures the rigidity, or 

 power of resisting distortion, of the medium. All such elastic 

 properties of the body as the velocity of propagation of waves 

 in it must evidently depend on the ratio n/p. 



Among the referees of one of Navier's papers was Augustine 

 Louis Cauchy (b. 1789, d. 1857), one of the greatest analysts of 

 the nineteenth century,* who, becoming interested in the 

 question, published in 1828f a discussion of it from an entirely 

 different point of view. Instead of assuming, as Navier had 

 done, that the medium is an aggregate of point-centres of force, 

 and thus involving himself in doubtful molecular hypotheses, 

 he devised a method of directly studying the elastic properties 

 of matter in bulk, and by its means showed that the vibrations 

 of an isotropic solid are determined by the equation 



8 2 e ( 1 4 \ 

 p = - [fc + -n\ grad div e - n curl curl e ; 



here n denotes, as before, the constant of rigidity; and the 

 constant &, which is called the modulus of compression^. denotes 

 the ratio of a pressure to the cubical compression produced by 

 it. Cauchy's equation evidently differs from Navier's in that 



* Hamilton's opinion, written in 1833, is worth repeating : " The principal 

 theories of algebraical analysis (under which I include Calculi) require to he 

 entirely remodelled ; and Cauchy has done much already for this great object. 

 Poisson also has done much ; but he does not seem to me to have nearly so logical a 

 mind as Cauchy, great as his talents and clearness are ; and both are in my 

 judgment very far inferior to Fourier, whom I place at the head of the French 

 School of Mathematical Philosophy, even above Lagrange and Laplace, though I 

 rank their talents above those of Cauchy and Poisson." (Life of Sir W. It. 

 Hamilton, ii, p. 58.) 



t Cauchy, Exercices de Mathematiques iii, p. 160 (1828). 



J This notation was introduced at a later period, but is used here in order to 

 avoid subsequent changes. 



