./. W. Gtbbs-~Equilibriurn of Jleterogeneous ISubstancts. 189 



second series differing infinitely little from the pair in the first, and 

 vice versa^ therefore the second series of coexistent phases must be 

 terminated by a critical phase wliic^h differs, but differs infinitely 

 little, from the first. We see, therefore, that if we vary arbitrarily 

 the values of any n — 1 of the quantities <,^>», /^ j, /^g? • • • Hn-, ii« deter- 

 mined by a critical phase, we obtain one and only one critical phase 

 for each set of varied values ; i. c., a critical phase is capable of 

 w— 1 independent variations. 



The quantities t,]>, //j, //g, • • . /^„, have the same values in two 

 coexistent phases, but the ratios of the quantities ^/, w, m,, rti,^^. . . m„, 

 are in general different in the two j)hases. Or, if for convenience we 

 compare equal volumes of the two phases (which involves no loss of 

 generality), the quantities //, mj, mg, , , . m„ will in general have 

 different values in two coexistent phases. Aj)plying this to coexis- 

 tent phases indefinitely near to a critical phase, we see that in the 

 immediate vicinity of a critical phase, if the values of n of the quanti- 

 ties t, J), /u^, //g? • • • Mn, iii'ti regarded as constant (as well as v), the 

 variations of either of the others will be infinitely small compared 

 with the variations of the quantities ?;, m^, rn^, . . . m„. This con- 

 dition, which we may write in the form 



(-1^) =0, (200) 



characterizes, as we have seen on page 171, the limits which divide 

 stable from unstable phases in respect to continuous changes. 



In fact, if we give to the quantities t, /j^, yUg, . . . yw„_i constant 



values determined by a i)air of coexistent phases, and to * a series 



of values increasing from the less to the greater of the values which it 

 has in these coexistent phases, we determine a linear series of phases 

 connecting the coexistent phases, in some part of which yu„ — since it 

 has the same value in the two coexistent phases, but not a uniform 

 value throughout the series (for if it had, which is theoretically im- 

 probable, all these phases would be coexistent) — must be a decreasing 



function of ", or of m„, if v also is sujjposed constant. Therefore, 



the series must contain phases which are unstable in respect to con- 

 tinuous changes. (See page 168.) And as such a pair of coexistent 

 phases may be taken indefinitely near to any critical phase, the 

 unstable jdiases (with resi)ect to continuous changes) must approach 

 indefinitely near to this phase. 



