J. W. Gihbs — Equilibrivni of Heterogeneous Substances. 369 



In expanding the product of the three sums, we may cancel on 

 account of the sign ^'' tlie terms which do not contain all the three 



3-3 



expressions dx^ dy, and dz. Hence we may write 



^, y., y. /dx dx dy dy dz dz \ 



3-3 3+3 \dx' dx' dj' dy' dz' dz'/ 



y { dx dy dz , /dx dy dz\) 



3+3 i dx' dy' dz' 3-3\dx' dy' dz' J ) 



„ /dx dy dz\ , /dx dy dz \ 



A-sXdx' dy' dz'/ 3-3\dx' dy' dz' J' ^ 



Or, if we set 



H= 



(437) 



Ave shall have 



(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 difterent values of r'^, which we may 

 denote by r^^, rg^, '/"g^, 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 1\^, i-^^, /"g^, \dz., 





\ (439) 



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

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

 function of ?/y, and the quantities E, F, G (or H), which we have 



1 . n dx. dz ^^. . . , ■, -, n 



expressed m terms oi y^, , . . . -^, . omce Sy, is a single-valued func- 

 tion of 7/v, and r^ ^, r.,^, r.^" (with respect to all the changes of which 

 the body is capable), and a symmetrical function with respect to rj^^ 

 rg-, ^'a^, and since r,^, r^-, r^"^ are collectively determined without 

 ambiguity by the values of E, F, and H, the quantity fy* niust be a 



