EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 205 



- < 418 > 



The determination of the fundamental equation for a solid is thus 

 reduced to the determination of the relation between e v /, 7/ V '> A, B, C, 

 a, b, c, or of the relation between \/^ T , t, A, B, C y a, b, c. 



In the case of isotropic solids, the state of strain of an element, so 

 far as it can affect the relation of e v , and TJ T) or of \fs v > and t, is capable 

 of only three independent variations. This appears most distinctly 

 as a consequence of the proposition that for any given strain of an 

 element there are three lines in the element which are at right angles 

 to one another both in its unstrained and in its strained state. If 

 the unstrained element is isotropic, the ratios of elongation for these 

 three lines must with IJ T determine the value v >, or with t determine 

 the value of \fs v >. 



To demonstrate the existence of such lines, which are called the 

 principal axes of strain, and to find the relations of the elongations 



fine dz 



of such lines to the quantities -j,, . . . -T-,, we may proceed as follows. 



The ratio of elongation r of any line of which a', /3', y are the 

 direction-cosines in the state of reference is evidently given by the 



equation 



dx , dx ,dx A 2 



dz , . dz 



Now the proposition to be established is evidently equivalent to this 

 that it is always possible to give such directions to the two systems 

 of rectangular axes X', Y', Z ', and X, Y, Z, that 



(421) 



^ _ _ 



dx' dx'~ ' dy'~ 



We may choose a line in the element for which the value of r is at 

 least as great as for any other, and make the axes of X and X' parallel 

 to this line in the strained and unstrained states respectively. 



Then = = 



