168 



BUTLER 



ART, D 



columns of (245) and (247). But in this case it is more con- 

 venient to make dm = 0. Then we may write 



dt dt 



dt = -r dr] -\- — dv, 

 dti dv 



dp dp 



dp = ~r dr) -\- — dv; 

 dr] dv 



and, when dt = 0, the value of dp/dv is given by 



dH 



(249) 



drf 



dh 

 dvdrj 



dr]dv 



dh 

 dv"" 



(250) 



since, by (44), t = {dt/dr))^^^^ and p = — (c?e/dy);^,„. In 

 stable phases, {dp/dv)i^^ must be negative. Thus, expanding 

 (250), a phase of invariable composition is stable when 



d^e dh / dh ' 

 drf^ dv^ \drjdv 



J > 0, 



dh 



d;;^>'- 



(251) 



The physical meaning of these conditions can be seen from a 

 consideration of the rj-v-e surface for homogeneous phases. Let 

 rj, V, € be the coordinates on this surface of the point D, rep- 

 resenting the phase in question. Let E be the neighbouring 

 point on the surface, with coordinates rj + Arj, v -{- Av, e -\- Ae, 

 and E' the point of intersection of the tangent plane through D 

 with the vertical erected at E. (See Fig. 6.) Let the ordinate 

 of E' he € -{- Ae'. Then, to the second order of small quantities, 





