364 J. W. Gibbs — Eqidlibriuni of Heterogeneous Substances. 



function of the temperatnre and the state of strain, so tliat for con- 

 stant temperature we might write 



6Yv,=i-i'(x„,j;|), 



the quantities X^,, . . . Z^, being defined by this equation. This 

 would be only a formal change in the definition of Xy^, , . . . Z^, and 

 would not affect their values, for this equation holds trixe of JC^, , 

 . . . Zjj as defined by ecination (355). With such data, by transfor- 

 mations similar to those which we have employed, we might obtain 

 similar results.* It is evident that the only difference in the equa- 

 tions would be that il\, would take the place of e^, , and that the 

 terms relating to entropy w^ould be wanting. Such a method is 

 evidently preferal)le with respect to the directness with which the 

 results are obtained. The method of this paper shows more distinctly 

 the role of energy and entropy in the theory of equilibrium, and can 

 be extended more natfti-ally 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 145.) 



We have already had occasion to remark that the state of strain 

 of any element considered without refei-ence to directions in space is 

 capable of only six independent variations. Hence, it must be possi- 

 ble to express the state of strain of an element by six functions of 



dot dz 



— — , , . . . -^, , which are independent of the position of the element. 



ax az 



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 quanti- 

 ties by ^, ^, C, a, b, c, we shall have 



* For an example of tliis method, see Thomson and Tait's Natural Philosopliy, 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 of 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. 



