Page:Scientific Papers of Josiah Willard Gibbs.djvu/239

between the quantities denoted by $\epsilon _{V'},\eta _{V'},{\frac {dx}{dx'}},...{\frac {dz}{dz'}}$, involving also the quantities which express the composition of the body, when that is capable of continuous variation, or any other equation from which the same relations may be deduced, may be called a fundamental equation for that kind of solid. It will be observed that the sense in which this term is here used, is entirely analogous to that in which we have already applied the term to fluids and solids which are subject only to hydrostatic pressure.

When the fundamental equation between $\epsilon _{V'},\eta _{V'},{\frac {dx}{dx'}},...{\frac {dz}{dz'}}$ is known, we may obtain by differentiation the values of $t,X_{X'},...Z_{Z'}$ in terms of the former quantities, which will give eleven independent relations between the twenty-one quantities

$\epsilon _{V'},\eta _{V'},{\frac {dx}{dx'}},...{\frac {dz}{dz'}},t,X_{X'},...Z_{Z'},$

(415)

which are all that exist, since ten of these quantities are independent. All these equations may also involve variables which express the composition of the body, when that is capable of continuous variation.

If we use the symbol $\psi _{V'}$ to denote the value of $\psi$ (as defined on page 89) for any element of a solid divided by the volume of the element in the state of reference, we shall have

$\psi _{V'}=\epsilon _{V'}-t\eta _{V'}.$

(416)

The equation (356) may be reduced to the form

$\delta \psi _{V'}=-\eta _{V'}\delta t+\textstyle \sum \sum '\displaystyle \left(X_{X'}\delta {\frac {dx}{dx'}}\right)\cdot$

(417)

Therefore, if we know the value of $\psi _{V'}$ in terms of the variables $t,{\frac {dx}{dx'}},...{\frac {dz}{dz'}},$ together with those which express the composition of the body, we may obtain by differentiation the values of $\eta _{V'},X_{X'},...Z_{Z'},$ in terms of the same variables. This will make eleven independent relations between the same quantities as before, except that we shall have $\psi _{V'}$ instead of $\epsilon _{V'}$. Or if we eliminate $\psi _{V'}$ by means of equation (416), we shall obtain eleven independent equations between the quantities in (415) and those which express the composition of the body. An equation, therefore, which determines the value of $\psi _{V'}$ as a function of the quantities $t,{\frac {dx}{dx'}},...{\frac {dz}{dz'}}$ and the quantities which express the composition of the body when it is capable of continuous variation, is a fundamental equation for the kind of solid to which it relates.

In the discussion of the conditions of equilibrium of a solid, we might have started with the principle that it is necessary and sufficient