Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/170

his theory, in the form to which it had then been brought, as a final explanation of the properties of light. "If we are asked," he wrote, "what reasons can be assigned for the hypotheses on which the preceding theory is founded, we are far from being able to give a satisfactory answer. We are obliged to confess that, with the exception of the law of vis viva, the hypotheses are nothing more than fortunate conjectures. These conjectures are very probably right, since they have led to elegant laws which are fully borne out by experiments; but this is all we can assert respecting them. We cannot attempt to deduce them from first principles; because, in the theory of light, such principles are still to be sought for. It is certain, indeed, that light is produced by undulations, propagated, with transversal vibrations, through a highly elastic aether; but the constitution of this aether, and the laws of its connexion (if it has any connexion) with the particles of bodies, are utterly unknown."

The needful reformation of the elastic-solid theory of reflexion was effected by Green, in a paper^[1] read to the Cambridge Philosophical Society in December, 1837. Green, though inferior to Cauchy as an analyst, was his superior in physical insight; instead of designing boundary-equations for the express purpose of yielding Fresnel's sine and tangent formulae, he set to work to determine the conditions which are actually satisfied at the interfaces of real elastic solids, These he obtained by means of general dynamical principles. In an isotropic medium which is strained, the potential energy per unit volume due to the state of stress is

$\phi ={\frac {1}{2}}\left(k+{\frac {4}{3}}n\right)\left({\frac {\partial e_{x}}{\partial x}}+{\frac {\partial e_{y}}{\partial y}}+{\frac {\partial e_{z}}{\partial z}}\right)^{2}+{\frac {1}{2}}n\left\{\left({\frac {\partial e_{z}}{\partial y}}+{\frac {\partial e_{y}}{\partial z}}\right)^{2}+\left({\frac {\partial e_{x}}{\partial z}}+{\frac {\partial e_{z}}{\partial x}}\right)^{2}\right.$

$\left.+\left({\frac {\partial e_{y}}{\partial x}}+{\frac {\partial e_{x}}{\partial y}}\right)^{2}-4{\frac {\partial e_{y}}{\partial y}}{\frac {\partial e_{z}}{\partial z}}-4{\frac {\partial e_{z}}{\partial z}}{\frac {\partial e_{x}}{\partial x}}-4{\frac {\partial e_{x}}{\partial x}}{\frac {\partial e_{y}}{\partial y}}\right\}$,

where e denotes the displacement, and k and n denote the two

1. Trans. Camb. Phil. Soc., 1838; Green's Math. Papers, p. 245.