FUNDAMENTAL EQUATIONS OF IDEAL GASES 361 

 using equation (VII). Finally the equation for e becomes 



.. = ™, |_ c..d(- ^-3^ (^) + ».£.. (62) 



This equation is, as it should be, identical with the energy 



/de\ 

 equation obtained by starting directly with the ( — j differential 



equation. 



The entropy may be computed by solving the equations 



©. = &).• @). =r ^^^^ 



The entropy expression, using (VII) in connection with the 

 first differential equation becomes, after adding and subtracting 



V — Bitrhi . 



aiTUi log ' 



mi 





V — Bitrii 



+ m,/i(0 + m,Hx. (64) 



Integration gives finally 



m = rmMt) + a^m, log ^^^ - ^^T^^) Yt + ''''^'- ^^^^ 



Starting with the second differential equation there results, 

 again using (VII), 



•ni = wi 



I 1 dt -\- mifiiv) + miHi = nii j ci* — 

 + mi / / ti— 1 fit' y + mi/i(t;) + mj/fi 



= mi / ci* y + / f — j - y dy + mi/i(i;) + mi^i 



P dt ai miH dBi , ^ , ,^^, 



= ^^ 1 ''* 7 - (. - 5imi) ~^ + ^^-^^^^^ + ^'^^- ^^^^ 



