The work of Green proved a stimulus not only to MacCullagh but to Cauchy, who now (1839) published yet a third theory of reflexion.[1] This appears to have owed its origin to a remark of Green's,[2] that the longitudinal wave might be avoided in either of two ways—namely, by supposing its velocity to be indefinitely great or indefinitely small. Green curtly dismissed the latter alternative and adopted the former, on the ground that the equilibrium of the medium would be unstable if its compressibility were negative (as it must be if the velocity of longitudinal waves is to vanish). Cauchy, without attempting to meet Green's objection, took up the study of a medium whose elastic constants are connected by the equation
,
so that the longitudinal vibrations have zero velocity; and showed that if the aethereal vibrations are supposed to be executed at right angles to the plane of polarization, and if the rigidity of the aether is assumed to be the same in all media, a ray which is reflected will obey the sine-law and tangent-law of Fresnel. The boundary-conditions which he adopted in order to obtain this result were the continuity of the displacement e and of its derivate , where the axis of x is taken at right angles to the interface.[3] These are not the true boundary-conditions for general elastic solids; but in the particular case now under discussion, where the rigidity is the same in the two media, they yield the same equations as the conditions correctly given by Green.
The aether of Cauchy's third theory of reflexion is well worthy of some further study. It is generally known as the contractile or labile[4] aether, the names being due to William
- ↑ Comptes Rendus, ix, p. 676 (25 Nov., 1839), and p. 726 (2 Dec., 1839).
- ↑ Green's Math. Papers, p. 246.
- ↑ Comptes Rendus, x, p. 347 (March 2, 1840): xxvii, p. 621 (1848); xxviii,p. 25 (1849). Mém. de l'Acad., xxii (1848), pp. 17, 29.
- ↑ Labile or neutral is a term used of such equilibrium as that of a rigid body on a perfectly smooth horizontal plane.
