534 RICE 



ART. L 



the calculus due to Euler, which states that if ^{x, y, z, . . .) 

 is a homogeneous function of the q^^ degree in its variables then 



d\p dyj/ d\J/ 



dx dy dz 



As a special case of this we see that 



9<^ d(j) d<i> d4> d<t> 



07] dv drrii 9w2 dnin 



But by the fundamental differential equation [86] which 

 expresses the conditions of equilibrium 



90 90 90 



Hence 



e = tt] — pv + fxinii + M2W2 . . . + iJLnm„ , 

 which is equation [93]. 



13. Linear Functional Relations in Surface Phases 



Precisely similar arguments justify equation [502], since we 

 assume as an obvious physical fact that if we consider two 

 surfaces of discontinuity of exactly similar constitution then 

 the entropy, energy, and amounts of the several components in 

 each would be proportional to the superficial extent of each. 

 Since e-^ is homogeneous of the first degree in the variables 

 7]^, s, nii^, W2'5, etc., it follows that the partial differential coeffi- 

 cients of the function 4>{'r]S, s, mr^, m2^, . . .) of these variables, 

 which is equal to e'^, with regard to the variables are individu- 

 ally also homogeneous functions of the variables of degree 

 zero, i.e., they are functions of the ratios of these variables. 

 But by [497] 



90 90 90 



^ = :97^' ^ = 7.' ^^ = ^' ^^'- (^2) 



Hence the n -\- 2 quantities t, a, \i\, 1JL2, ... are functions of 



