Steam of Maximum Density, 173 



given temperature. When the intervals taken are small (as one 

 degree Centigrade), such mean value coincides exactly, so far as 

 observations are concerned, with the rate of increment per degree 

 of the force existing at the middle point of the interval or at the 

 given temperature. Such rate of increment per degree is the 



AP 



hyperbolic logarithm a taken to represent -p-« 



According to the known properties of hyperbolic logarithms, 

 when the rate of increment of force is constant throughout any 

 interval (as one degree taken as a unit), and when such interval is 

 divided into an infinite number (q) of equal parts, then we have 

 for the constant ratio of increase of force for such infinitely small 



interval (l + -) = e?, and consequently for the total unit inter- 

 val containing q such parts, 



(l + -Y = ef x? =e« = l + r, 



if it is known that the force unity under a constant rate of incre- 

 ment becomes (1 + r) at the end of the unit interval. From the 

 above equation we obtain the following two values of « in terms 

 of r : — 



a= com log (1 +r) x - f 



K 



and, by expressing hyperbolic logarithm in terms of its corre- 

 sponding number, 



a=r -2 + W- kc - 

 In the above equations the indefinitely small quantity - represents 



, , p dV AP 1 

 ^.logP=- F = T -x-. 



AP 



The values of -=-, or the rates of increment of force P for 



every tenth degree of temperature, commencing with 30° C. 

 below the freezing-point of water, have been obtained in the 

 manner above described from M. Regnault's principal and 

 adopted Table, and are contained in the last column of Table I. 



hereunto annexed. The ratios of these quantities I ■=- \ to one 



another are the same (or very nearly the same), at the same tem- 

 peratures, as the ratios of the differentials (d. log P). On com- 



AP 



paring together these ratios -p- , it is found that they all bear a 



