SURFACES OF DISCONTINUITY 607 



and its minor determinants. Thus if the form is to be definitely 

 positive, this determinant, the first minors obtained by erasing 

 any row and a corresponding column, the second minors ob- 

 tained by erasing any two rows and the corresponding columns, 

 the third minors obtained in a similar way, and so on until we 

 reach the individual constituents of the leading diagonal, must 

 all be positive quantities. If this is not so the form will have 

 negative values for some sets of values of ^i, ^2, ^3, ... and so 

 the system will for some displacements not tend to return to, 

 but will move further away from, the original equilibrium con- 

 figuration. Indeed if the first minors, third minors, fifth minors 

 and so on had one sign; the determinant, the second minors, 

 the fourth minors and so on, the other; the system would be 

 unstable for any displacement whatever. 



39. Restricted Character of such Conditions as Applied to a 



Thermodynamical System 



In the investigation of the stability of a thermodynamic 

 system a similar procedure can be followed, but it suffers from 

 one limitation which Gibbs discusses. The energy of the 

 system is regarded as a function of the thermodynamical 

 variables, which in the present instance specify the condition of 

 the homogeneous masses and of the film separating them. For 

 equilibrium 6e must be zero for any arbitrary infinitesimal 

 variations of these variables — ^at least, arbitrary apart from the 

 familiar conditions such as [481].* For stable equilibrium Ae 

 will be positive for all possible variations of the variables within 

 the assigned limitations. If we then proceed to apply the 

 method just outlined we must conceive e to be formulated as 

 a function of the variables, (the entropy, masses of components, 

 volume, area of film) and the first and higher differential coeffi- 

 cients also so expressed and the tests applied. (See the proof 

 for the thermodynamic system as given on pages 105-115, 

 especially [173] et seq.) But this assumes that in any state, 

 other than the initial one, whose energy content needs to be 



* This restriction in arbitrariness would render the analytical pro- 

 cedure in such a case somewhat more complicated than that indicated 

 above, but would not invalidate the general idea. 



