And 



and 



the Second Law of Thermodynamics* 371 



***t**-~!~* ( 3 ) 



In order that this may represent a complete differential, 



m(-7 -j—jdv + -^p dlog T must be a complete differential 



of some function of v and T. Let u be this function, then 



1 /dU dU'\ M 1 dU _ 



du ■ du ..n 

 --ch* dv+ dY dT; 



dw 1/dU dU'\ 

 *'• dv T\dv dv)' 







and 





du 1 dU . 





<iT""TrfT' 





and as 





tZ cZm d 4 





dTdv~-"dvdT> 





T\dT dv dT dv/ T 2 Vdv "" dv)~Tdi 

 1/^U_^IP\ _d_dU 

 TVdv dv/ dT dv 



dU 

 dv dT 



i^du_dip^ = _^di7 (5y 



which is the condition necessary to make -«r a complete dif- 

 ferential. 

 Then 



^=dlogT + fdlogv-^^dv+^dlogT (6) 



= d(logT + flogv + «) .(f) 



If ; however, such a relation should exist between the volume 



* The original determination of this necessary condition of any system 

 to which the second law will apply is due to Mr. S. H. Burbury. In the 



dU . dv dU' fa 



notation of M. Boltzmann j- is represented by V- ; and -r- by -£*, 



202 



