SURFACES OF DISCONTINUITY 621 



given value of v' (or v"; v' + v" is constant). So s is a single- 

 valued function of v', and ds/dv' has a definite meaning. We 

 can obtain n — 1 similar equations 



(.'-." + r,^).' + (/£ + ."^' 



araX 



+ s — 5mi + etc. = 0, 



etc. 



These n equations give us the theoretical means to calculate 

 the n quantities d\i\ldv\ dyti/dv' , ... in terms of the state of the 

 system. In this way we see, as is stated at the top of page 250, 

 that all the quantities relating to the system may be regarded as 

 functions of v'. Thus we can obtain d-p' /dv'; for it is equal to 



dux dv' ^ dti.dv' '^ • • • " ^' dv' "^ ^'' dv' 



Similarly 



dy" „djii . „dji2 



dv' - ''' dv' + ^^ dv'^ ■- 



and 



da djii dyii 



d^' ^ ~ ^'d? ~ ^'d^' ~ ••• 



In the initial state we assume that p' — p" = o-(ci + C2); 

 in the varied state the pressures and surface tension p' + 8p', 

 p" + bp", (J -{- b(T are of course the same functions of t, 

 Ml + ^Mi, ... as p', p", a are of t, ni, ... But nowhere 

 do we have to assume that 



(p' + bp') - ip" + Sp") = (<r + 8a) (ci + dci + c, + 8c,), 



so that the varied state need not he a state of equilibrium as regards 

 the condition expressed by equation [500]. 



The energy of the system, depending as it does on the variables 

 of the system, can, as we have just seen, be expressed as a func- 

 tion of v'. The energy in the varied state is by Taylor's 

 theorem 



