THERMODYNAMIC AL SYSTEM OF GIBBS 89 



Ae = tA-q — pAv + niArtii + HiArrh . . . + UnAtUn , 



where A77, Av, Ami, etc., are all proportional to the values of 

 7], V, mi, etc. in the original mass. We may thus continue these 

 additions until we have doubled the amount of the original 

 mass. Then, since At; = t], Av = v, Ami = mi, etc., the energy 

 of the added substance is 



Ae = It] — pv + iumi + nim^ . . . + m»w„ , 



and this must be equal to the energy t, of the mass originally 

 present. 



The general justification of this treatment depends on Euler's 

 theorem on homogeneous functions. According to this theorem, 

 a y = <f)(xi, X2,...Xn) be a homogeneous function of xi, 

 X2,. . .a:„ of the w"* degree; 



dy dy dy 



Xi — -i- X2— ... + x„ 7- = my. (52) 



0X1 00:2 OXn 



Now a homogeneous function of the w"" degree is one for which 



<j){kxi, kx2, . . . kx„) = k'"<t>{xi, X2, . . . Xn), 



i.e., if each variable Xi, X2,- . .Xn is multiplied by a quantity k, 

 the value of the function is multiplied by /b". The energy of a 

 homogeneous mass is evidently a homogeneous function of the 

 first degree with respect to 77, v, mi, m^,. . .m„. If we increase 

 each of these quantities k times, i.e., by taking k times as much of 

 the homogeneous substance, the energy is increased in the 

 same proportion. Therefore by Euler's theorem, putting 

 € = <f>(j], V, mi,. . .w„) we have 



de dt de de 



i = VT-i-v— +mi- — ... +m„ t — > 

 drj dv dmi dmn 



or 



t = r]t — vp -\- mi/xi . . . + mnUn, 



