248 APPENDIX B. 



and from equations (3), (4), 



eF'tdt = N \ogvdt; 

 or, dividing by F'tdt, 



N 

 e= -j^\ogv = Tlogv. 



JV 

 Calling T the fraction -^ which is a function of t 



only, the equation 



e = T log v 



is the analytical expression of the law stated pp. 80, 

 81. It is common to all gases, since the laws ot 

 which we have made use are common to all. 



If we call s the quantity of heat necessary to 

 change the air that we have employed from the 

 volume 1 and from the temperature zero to the 

 volume v and to the temperature t, the difference 

 between s and e will be the quantity of heat re- 

 quired to bring the air at the volume 1 from zero 

 to t. This quantity depends on t alone; we will 

 call it U. It will be any function whatever of t. 

 We shall have 



s = e + U= Tlogv + U. 



If we differentiate this equation with relation to t 

 alone, and if we represent it by T' and U', the dif- 

 ferential coefficients of T and U, we shall get 



//<? 



g=ZMogt;+Z7'; ... (5) 



