204 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



for equilibrium that the temperature shall be uniform throughout the 

 whole mass in question, and that the variation of the force-function 

 (-i/r) 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 \fs 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 



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

 would be only a formal change in the definition of X^>, . . . Z% and 

 would not affect their values, for this equation holds true of JT 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.* It is evident that the only difference in the equations would 

 be that i//v would take the place of e T , 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 rdle 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 



-T-7, . . . -j-,, which are independent of the position of the element. 



Ct/OC Ct/2/ 



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, 6, c we shall have 



* 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. 



