SURFACES OF DISCONTINUITY 



589 



This gives us one equation between the variables and so we 

 can reduce them from w + 3 in number to n + 2; the most 

 convenient set of variables is then r/, v, s, n, Vi, . . . r„_„ where 

 7-j = mi/nin, n = nii/nin, etc. So we write the function for c as 

 i{rj. V, s, n, r2, . . . ) and 



de{-n, V, s, ri r2, . . . ) = tdr{ — pdv + ads + vi dn + va drj + . . . , 



where vi, v-z, etc. are functions of r], v, s, n, r^, ... 

 The other three Gibbs functions are then 



yP{t, V, S, ri, rg, . . .) = e - tr], 

 ^(t, p, s, n, rz, . . .) = e - tr] + pv, 

 xiv, P, s, n, rg, ...) = e + pv, 

 with the differential equations 



d\l/{t, V, s, n, r2, . . .) = —vdi — pdv 



+ ods ■\- vidn -\- . . . , 

 d^{t, p, s, n, ra, . . .) = -vdt -{- vdp 



+ <^ds -\- vidri -{■ . . . , 

 dxiv, P, s, n, ra, . . .) = tdri + vdp 

 + ads -\- vidri -\- . . . 



From the second of those we have 

 9f («, P, s, ri, ra, ...) 



dt 



= - iC^, P, s, n, ra, ...) 



and 



d^{t, p, s, n, ra, ...) 

 ds 



By cross differentiation 

 dr]{t, p, s, ri, ra, . 



ds 



a{t, p, s, n, ra, ...)*. 



dajt, p, s, n, ra, ■ ..) 

 dt 



(22) 



Actually <r is only dependent on t, p and n — 2 of the ratios n, ri, . 



