§ 1. The general problem in elasticity, as usually presented for solution, supposes the elastic substance to pass initially from a state without strain. But important cases exist where it must be conceived to start from a state already under considerable stress. When this is the case, the constitution of the solid undergoes great change, as is shown by the fact that strained glass loses its isotropic property, and becomes doubly refractive. This subject was long ago attacked by Cauchy, who, by means of the theory of molecular actions, deduced the existence, in the expressions for the stresses due to the secondary strains, of terms proportional to the initial or primary stresses. The problem has been since discussed by MM. De St Venant and Boussinesq, who have applied to it Green's expression for the energy stored up during the strain. But the question, in their hands, still retains traces of Cauchy's hypothetical element, inasmuch as their expression for the potential energy was deduced by means of the molecular theory. M. De St Venant even considers it a strong argument for the truth of the latter, that it is indispensable in the discussion of this problem. These authors have also failed to see in what way the remaining part of the potential depends on the original strain.