SURFACES OF DISCONTINUITY 533 



further insight into the argument presented by Gibbs on page 

 226. Before leaving this topic, however, it may be as well to 

 enjoin on the reader the necessity of keeping Gibbs' own 

 caution in mind that in strict theory it is only for this specially 

 chosen position of the dividing surface that the equation [500] 

 is valid, and that only to it may the term surface of tension be 

 correctly applied. 



VI. The Adsorption Equation 



IS. Linear Functional Relations in Volume Phases 



Let us revert for a moment to the substance of pages 85-87 

 of Gibbs, which leads to the equation [93]. Divested as far 

 as possible of the mathematical dressing, the simple physical 

 fact on which it rests is this. We are considering two homo- 

 geneous masses identical in constitution and differing only in 

 the volume which they occupy. If the volume of the first mass 

 is r times that of the second, then the amount of a given constit- 

 uent in the first is r times that of the same constituent in the 

 second; also the energy and entropy of the first are respectively r 

 times the energy and entropy of the second. Hence, when we 

 express e as a function of the variables -q, v, mi, m^, ... w„, 

 writing for example, 



e = <^(r?, V, mi, m2, ... m„), 

 we know that 



(f){rr], rv, rmi, rm2, . . . rmn) = r4){y}, v, mi, W2, . . . w„). 



In other words, the function (/> is a homogeneous function of the 

 first degree in its variables.* There is a well-known theorem of 



* It should be observed that this does not of necessity mean a linear 

 function. Thus ax + by + cz is a linear function of the variables x, y, z; 

 but 



ax^ + fcy^ + cz^ 

 Ix + rny + nz 



is not. Yet both are homogeneous functions of the first degree; for if 

 I, y, z are all altered in the same ratio, the values of these functions are 

 also altered in the same ratio. 



