Page:Scientific Papers of Josiah Willard Gibbs.djvu/240

for equilibrium that the temperature shall be uniform throughout the whole mass in question, and that the variation of the force-function ($-\psi$) of the same mass shall be null or negative for any variation in the state of the mass not affecting its temperature. We might have assumed that the value of $\psi$ for any same element of the solid is a function of the temperature and the state of strain, so that for constant temperature we might write

$\delta \psi _{V'}=\textstyle \sum \sum '\displaystyle \left(X_{X'}\delta {\frac {dx}{dx'}}\right),$

the quantities $X_{X'},...Z_{Z'}$, being defined by this equation. This would be only a formal change in the definition of $X_{X'},...Z_{Z'}$ and would not affect their values, for this equation holds true of $X_{X'},...Z_{Z'}$ as defined by equation (355). With such data, by transformations similar to those which we have employed, we might obtain similar results.^[1] It is evident that the only difference in the equations would be that $\psi _{V'}$ would take the place of $\epsilon _{V'}$, and that the terms relating to entropy would be wanting. Such a method is evidently preferable with respect to the directness with which the results are obtained. The method of this paper shows more distinctly the rôle of energy and entropy in the theory of equilibrium, and can be extended more naturally to those dynamical problems in which motions take place under the condition of constancy of entropy of the elements of a solid (as when vibrations are propagated through a solid), just as the other method can be more naturally extended to dynamical problems in which the temperature is constant. (See ,note on page 90.)

We have already had occasion to remark that the state of strain of any element considered without reference to directions in space is capable of only six independent variations. Hence, it must be possible to express the state of strain of an element by six functions of ${\frac {dx}{dx'}},...{\frac {dz}{dz'}}$, which are independent of the position of the element. For these quantities we may choose the squares of the ratios of elongation of lines parallel to the three co-ordinate axes in the state of reference, and the products of the ratios of elongation for each pair of these lines multiplied by the cosine of the angle which they include in the variable state of the solid. If we denote these quantities by $A,B,C,a,b,c$ we shall have

1. For an example of this method, see Thomson and Tait's Natural Philosophy, vol. i, p. 705. With regard to the general theory of elastic solids, compare also Thomson's Memoir "On the Thermo-elastic and Thermo-magnetic Properties of Matter" in the Quarterly Journal oj Mathematics, vol. i, p. 57 (1855), and Green's memoirs on the propagation, reflection, and refraction of light in the Transactions of the Cambridge Philosophical Society, vol. vii.