90 



since 



BUTLER 



ART. D 



(-) 



\dv/v 

 xdmi/r,, V 



= t, 



m^ • • • mn 



= - P, 



7Jf TTli ' ' ' trifi 



wij - • • m-n 



= Hi, etc. 



(53) 



Euler's theorem further states that if e = 0(t/, v, m\, nh, . . .m„) 

 is a homogeneous function of the first degree 



9e 9e 



a^ "^' m; " ~ ^' 



be 

 drrii 



= )U], etc., 



are functions of zero degree. Therefore, applying Euler's 

 theorem to one of these functions, e.g. to 9e/9mi, we have: 



326 a^e dh 



+ V — + mi :r~l + ^ 



a^e 



dmi • dr] 



dmi • dv 

 + mn 



dm-^ 



drill • dm^ 



dh 



dmi • drrin 



= 0. 



(54) 



or 



dt 



dp dfii dfXi dfin , . 



V Z~~ - V -r^ -{- mi -— -\- m2-~ ... + mn z =0. (55) 

 dmi dm-i dmi dmi dnii 



Therefore, in general, 



7]dt — vdp + midfjLi + m2dp,2 . . . + m„c?jun = 0. (56) [97] 



Gibbs obtains this equation by differentiating (48) in the most 

 general manner, viz., 



de = tdr] + rjdt — pdv — vdp + mdmi + midm 



. . . + Hndmn -\-mndHn, 



and comparing the result with (44), which is a complete differ- 

 ential. 



Equation (56) provides a relation between the variations of 

 the ?i + 2 quantities, t, p, m,. . .ju„, which define the state of 



