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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

$\rho {\frac {\partial ^{2}\mathbf {e} }{\partial t^{2}}}=-3n\ \mathrm {grad\ div} \ \mathbf {e} -n\ \mathrm {curl\ curl} \ \mathbf {e}$,

where n denotes a constant which measures the rigidity, or power of resisting distortion, of the mediun, All such elastic properties of the body as the velocity of propagation of waves in it must evidently depend on the ratio n/ρ.

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,^[1] who, becoming interested in the question, published in 1828^[2] 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

$\rho {\frac {\partial ^{2}\mathbf {e} }{\partial t^{2}}}=-\left(k+{\frac {4}{3}}n\right)\mathrm {grad\ div} \ \mathbf {e} -n\ \mathrm {curl\ curl} \ \mathbf {e}$;

here n denotes, as before, the constant of rigidity, and the constant k, which is called the modulus of compression,^[3] denotes the ratio of a pressure to the cubical compression produced by it. Cauchy's equation evidently differs from Navier's in that

1. Hamilton's opinion, written in 1833, is worth repeating: "The principal theories of algebraical analysis (under which I include Calculi) require to be entirely remodelled; and Cauchy has done much already for this great object. Poisson also has done much; but he does not seen 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, eren above Lagrange and Laplace, though I rank their talents above those of Cauchy and Poisson." (Life of Sir W. R. Hamilton, ii, p. 58.)
2. Cauchy, Exercices de Mathématiques iii, p. 160 (1828).
3. This notation was introduced at a later period, but is used here in order to avoid subsequent changes.