SURFACES OF DISCONTINUITY 537 



15. Variations and Differentials 

 The apparent formal similarity of equations [497] and [501] 

 should not blind the reader to the different implications of the 

 two, which the alternative method of writing the derivation of 

 [508] may help to bring out. In equation [497] the functional 

 dependence in the mathematical sense of e^ on the variables 

 rj-s, s, mi-s, W2'5, etc., is kept in the background as it were; 8e^, 

 dr}^, brrii^, 8m2^, etc., are any arbitrary infinitesimal variations of 

 t^, etc., in other words, although t^ is some function of the quan- 

 tities Tjs, s, rrix^, mi^, etc., presumably discoverable by experi- 

 ment, €^ -{■ 8e^ is not necessarily equal to this same function of 

 the quantities tjs + Srj^, s + 5s, nii^ + Snii^, rrbi^ -\- 8m2^, etc. ; 

 i.e., the varied state is not of necessity one of equilibrium. 

 Equation [497], while being the statement of the condition 

 that the unvaried state is one of equilibrium, is from the 

 mathematical point of view a way of writing down the n + 2 

 partial differential equations (12). But in [501] the quantities 

 dr]^, ds, drtii^, dnii^, etc. are not arbitrary variations but differ- 

 entials whose values must be chosen so that the varied state is 

 one of equilibrium as well as the initial, i.e., so that t^ + de^ 

 is the same function of ??« -f d-q^, s + ds, Wi^ + dmi^, nii^ + 

 d?r.2^, etc., as e^ is of t?^, s, mi^, m2«, etc. If this is kept in mind it 

 will be seen from the nature of the proof of [508] that, in passing 

 from any state of the system for which [508] is assumed to be 

 true to any other for which it is also true, we must pass through a 

 series of equihbrium states; briefly all the changes involved 

 must be reversible in the usual thermodynamic sense, not 

 merely in the special sense in which Gibbs uses that word. 

 More than one writer has pointed out that in some of the 

 operations carried out in certain experiments made to test the 

 validity of the adsorption equation this condition has apparently 

 not been satisfied and irreversible steps have intervened. 

 Further reference will be made to this presently, but it is this 

 feature of the proof to which we have drawn attention that is 

 involved. 



16. Condition for Experimental Tests 

 In many of the experiments made to test the truth of [508] the 

 adsorption is measured at the surface of bubbles of a gas or 



