EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



209 



Or, if we set 



(437) 



we shall have 



G = H*. (438) 



It will be observed that F represents the sum of the squares of the 

 nine minors which can be formed from the determinant in (437), and 

 that E represents the sum of the squares of the nine constituents of 

 the same determinant. 



Now we know by the theory of equations that equation (431) will 

 be satisfied in general by three different values of r 2 , which we may 

 denote by rf, r 2 2 , r 3 2 , and which must represent the squares of the 

 ratios of elongation for the three principal axes of strain; also that 

 E, F, G are symmetrical functions of r x 2 , r 2 2 , r 3 2 , viz., 



(439) 



Hence, although it is possible to solve equation (431) by the use of 

 trigonometrical functions, it will be more simple to regard T as a 

 function of JJ T and the quantities E, F, G (or H), which we have 



expressed in terms of -?-? , . . . -T-? . Since e v , is a single- valued function 

 of t] v and r^ y r 2 2 , r 3 2 (with respect to all the changes of which the 

 body is capable), and a symmetrical function with respect to 2 



r 



2 , 



r 3 2 , and since r x 2 , r 2 2 , r 3 2 are collectively determined without ambiguity 

 by the values of E, F, and H, the quantity e V ' must be a single- valued 

 function of j/ V '> E, F, and H. The determination of the fundamental 

 equation for isotropic bodies is therefore reduced to the determination 

 of this function, or (as appears from similar considerations) the deter- 

 mination of i/r v , as a function of t, E, F, and H. 



It appears from equations (439) that E represents the sum of the 

 squares of the ratios of elongation for the principal axes of strain, 

 that F represents the sum of the squares of the ratios of enlargement 

 for the three surfaces determined by these axes, and that G represents 

 the square of the ratio of enlargement of volume. Again, equation 

 (432) shows that E represents the sum of the squares of the ratios of 

 elongation for lines parallel to X', Y' } and Z' ; equation (434) shows 

 that F represents the sum of the squares of the ratios of enlargement 

 for surfaces parallel to the planes X'-Y', Y'-Z', Z'-X' '; and equation 



(438), like (439), shows that G represents the square of the ratio of 

 G. i. o 



