Abstract: This paper is neither laudatory nor derogatory but it simply contrasts with what might be called elastosiatic (or static topology), a proposition of the famous six equations. The extension strains and the shearing strains which were derived by A.L. Cauchy, are linearly expressed in terms of nine partial derivatives of the displacement function(u_i, u_j, u_k) =u(xⁱ, x^j, x^k) and it is impossible for the inverse proposition to sep up a system of the above six equations in expressing the nine components of matrix ( (u_i,u_j,u_k)/∂(xⁱ, x^j, x^k)). This is due to the fact that our geometrical representations of deformation at a given point are as yet incomplete[1]. On the other hand, in more geometrical language this theorem is not true to any triangle, except orthogonal, for "squared length" in space[2].The purpose of this paper is to describe some mathematic laws of algebraic elastodynamics and the relationships between the above-mentioned important questions.

Key words: shallow spherical shell, shallow conical shell, thermoelastically coupled, nonlinear vibration, perturbation method