Second Law of Thermodynamics, 577 



9. Again, since U = jJUF da da\ 



^U=rfuBF^aV+ ^F^Vdadcr', 

 = f ill dF da da' + (Tf2 ^By do- do', 



= rfuaF^^(7 / +s^d^ 



Therefore 



3U-2^aw=jyU3Fdo-^. . . . (II.) 



In the same way, since U=JU/do, we find , 



aU"-2^3w=Jua/rfo-. . . . (H. <o 



10. If BQ be the energy imparted to the system or spent 

 in work when the variation takes place, 



=BT + SU— 2*-r- "ftv + S-j- du, because -—=0, 

 av dv dt 



= 2dT+ g UdFtfodo'-jJTdFrfodo' 



by (I.) and (II.). Therefore 



^=231ogT+ tf^-d¥d<rd</- ff^Frfodo'. . (III.) 



11. We might now make -^- a complete differential by 



assuming F = ^>( — rp — J, where c/> is an arbitrary function. 



But this solution must be rejected, because it will not make 

 F vanish for all infinite values of r and U. 



12. But using (II. a) instead of (II.) we obtain instead of 



(in.) 



^=291ogT + 2f?d/^-ffl±5dFtf^<7'. . (IV.) 



