158 DYNAMICAL THEOKY OF SOUND 



substance can be completely mapped out. The other suc- 

 cessions referred to are those in which there is no gain or loss 

 of heat to the substance, as if it were enclosed in a vessel (o: 

 variable volume) whose walls are absolute non-conductors. The 

 corresponding lines are therefore called " adiabatics." 

 In a perfect gas we have 



p = R P 0, or pv = R0, ............... (1) 



where 6 is the absolute temperature on the gas thermometer 

 and R is a constant depending on the nature of the gas. The 

 isothermal lines pv = const, are therefore rectangular hyperbolas 

 asymptotic to the coordinate axes. As regards the adiabatics 

 the heat required to increase the pressure by &p when the 

 volume is constant will be given by an expression of the form 

 PSp. If c denote the specific heat (per unit mass) at constant 

 volume, this must be equal to c&0, where $0 is the corre- 

 sponding change of temperature. Now when $v = we have 

 Sp/p = $0/0, whence, comparing, P = c0/p. Again, the heat 

 required to' augment the volume by Iv when the pressure is 

 constant- may be denoted by QSv, which must be equal to c'&0 

 where c' is the specific heat at constant pressure. Since, when 

 fy = we have &v/v = $0/0, we find Q = c0/v. The heat ab- 

 sorbed when both pressure and volume are varied infinitesimally 

 is therefore 



............... (2) 



and the differential equation of the adiabatics is therefore 



$M* = 0. ...(3) 



p c v 



The ratio c'/c. of the two specific heats is practically constant. 

 Denoting it by 7, .we have 



\ogp 4- 7 log v = const., 

 or pv? = const., ........................ (4) 



as the equation of the adiabatic lines. The value of 7 as found 

 by direct experiment is about 1*41 for air, oxygen, nitrogen, 

 and hydrogen. The figure shews the isothermal and adiabatic 

 lines through a point P of the diagram, the latter curve being 

 the steeper. 



