SURFACES OF DISCONTINUITY 617 



the complete system is stable. Now if the complete system is 

 stable, the original system (without communication with 

 external mass) is certainly stable. For blocking up the tubes 

 and isolating the original system is equivalent to imposing a 

 mechanical constraint on the complete system; and it is well 

 known in mechanics that if a dynamical system is in a stable 

 state of equilibrium, the imposition of a constraint does not 

 upset that condition. Indeed this fact is intuitively obvious. 

 The inequahty [549] is simply the same result extended to a 

 wider system. But, of course, the condition may not be 

 necessary for stability of equilibrium as regards movement of the 

 surfaces; in short it insures stability for the system under wider 

 conditions than are actually envisaged at the outset and so 

 under more restricted conditions than these the system might be 

 stable without [549] being satisfied, 



43. Gibhs' General Argument Concerning Stability in Which the 

 Difficulty Referred to in Subsection {39) Is Surmounted 



The general argument of Gibbs on the conditions of stability 

 or instability will be found on pages 246-249, (On pages 

 242-246 he discusses the problem by a more specialized method 

 which can be passed by for the moment.) At the outset of the 

 argument he raises the point which we have already noted, that 

 if we use an anal3rtical method, analogous to that employed in 

 dynamics, we are virtually excluding from consideration those 

 states of the system which are not in equilibrium and for which 

 the fundamental equations are not valid and the usual func- 

 tional forms for energy, etc. have no meaning, since in these 

 states the systems cannot be specified with precision by values 

 of the usual variables. That is dealt with on page 247. He 

 proposes then to surmount this obstacle by introducing the 

 consideration of an "imaginary system" which is fully de- 

 scribed at the top of page 248. This system agrees with the 

 actual system in all particulars in the initial state, which is one 

 of equilibrium for both systems, though whether it is stable or 

 not for the actual system is the point under consideration. His 

 argument, however, may be framed so as to exclude any express 

 consideration of his imaginary system and may appear simpler 



