18 M. L. A. Colding on the Universal Powers of 



points m, m\ m", &c, and w represents the energy which the 

 fluid actually contains. 



Provided the fluid is a gas, we may without any appreciable 

 error leave out of the formula the terms which depend on the 

 mutual attraction of the particles of air, and then the formula 

 may be written 



■ q=G-w (35) 



When the mass of air by which the work is performed is sup- 

 posed to be = /x, then, conformably to formula (14) ^ we have, by 

 differentiation of the equation (35), 



ll— — — dw. 



p 



But, in accordance with Mariotte and Gay-Lussac's law, p is a 

 given function of p and #, determined by the equation 



p = kp(l+*0), (36) 



being the temperature Celsius, and a the coefficient of expan- 

 sion for the gas, and k the ratio between the pressure of air 

 gmh and the density D at 0°. Thus we find 



the whole integral of which is 



v,=f(6)-pk(l + aO)lo S £-, . . . (37) 

 Po 

 f(6) denoting an arbitrary function of 0, p any constant pressure, 

 and log represents the natural logarithm. 



Formula (37) is exactly the same as Holtzmann has derived for 

 vapour {of water)* by a procedure similar to that which was first 

 stated by Clapeyronf. 



As is well known, the specific heat at constant pressure is de- 

 rived from this formula, 



.= !>(«, -fafogA (38) 



f 1 Po 



and the specific heat at constant volume will be 



"z^lf'm+teH %-*■*«, ■ • • (39) 



in which f'(6) denotes the differential coefficient of f(6) with 

 regard to 9. From formulae (38) and (39) we may also easily 

 derive formula (20). 



If steam is at maximum of density, and the quantity of heat 

 w contained in the same mass of steam is supposed to be constant, 

 then, conformably to formula (37), the pressure is a function 



* Pogg. Ann. d. Physik, Erg'anzungsband ii. p. 183. 

 t Ibid. vol. lix. p. 446. 



