Nature and their Mutual Dependence. 11 



If this equation is integrated, and we observe that s is always 

 supposed to be very small, then we have without any appreciable 

 error 



= ?o+ ^jpy.* C 1 ?) 



supposing q=-q for 5=0. 



If the temperature of the gas in its original state of equili- 

 brium at the density D be denoted by T, and the temperature 

 during the compression at the instant in question be denoted by 

 (T + 6) } then, if the coefficient of expansion of the air is u, we have 



gmh l + a(T + 0) 



p-v-p 1+«T ' 



When we here substitute the values of p and p, according to 

 the formulas (15) and (16), we get without appreciable error 



S -(1+«T)( 7 -1)' 

 which, when substituted in formulas (17), gives 



gmh .y.uO 

 q ~ q ° + D(l + aT)( 7 -l/ 



If the density of the gas at 0° during the pressure gftih is 

 equated to D , then is 



D(1 + «T)=D , 

 and consequently we have 



If the velocity by which q varies in proportion to the temperature 

 is denoted by co, which represents the specific heat of the fluid 

 of variable volume, then we have 



. . . «-% (19) 



By differentiating the equation (18) with regard to 6, we then 

 get 



"^•#i"' ri 



and as this expression is not changed, however small s is sup- 

 posed to be, we perceive that formula (20) must represent the 

 exact expression for the specific heat of variable volume. 



When now we denote the specific heat of variable volume for 

 another gas by co', and the density at 0° under the pressure gmh 

 is denoted by D' , and the ratio between the specific heat of this 



