§6 OF ARTS AND SCIENCES. 377 



In this formula, if we suppose T constant, that is if 



= o, we nnd 

 dv 



y dP^_ p:_ ( \r-z T 



dV 3 V /— /■ 



+ 2 P^o\ 



hut the coefficient of resiHence, E^ being defined as the 

 hmit of the numerical ratio of the increment of external 



pressure i^P) to the corresponding voluminal compres- 

 sion, ( — ~^j, the temperature being constant, is equal alge- 



dl^ 



braicallv to the first term V : hence we have 



dV 



Now if in I. we suppose the pressure to be constant, that 



dP 

 is if — — = o, we shall have, transposing and inverting, 



dV P' 



^ III. 



since e is defined as the coefficient of voluminal expansion 

 under constant pressure. 



The truth of the formulae II. and III. is easily tested by 

 multiplying them together. We have 



Be = , 



a relation already established by mechanical considera- 

 tions (^2). 



/ ' 

 If, moreover, in III., we put -—- = o, and P -{- P' ^^ 



if 



