172 Mr. T. R. Edmonds on the Elastic Force of 



low to the very high temperatures. It will be readily seen that, 

 for short ranges of temperature, the successive values of A log P 

 are in near agreement with Dalton's law, and that for moderate 

 ranges the values of A log P are in near agreement with Roche's 

 law. The result of the inspection (if carried to a sufficient 

 degree of minuteness) can hardly fail to be the conviction that 

 there exists a fixed and simple law of connexion between the 

 force and temperature, and that Roche's law is only an approxi- 

 mation (and not a very close one) to such fixed and simple law. 



The manifest uniformity of decrease with temperature of the 

 finite differences of the logarithms of the observed elastic forces 

 of steam of maximum density is indicative of similar uniformity 

 of decrease of the differential or indefinitely small differences of 

 these logarithms. If the elastic force (P) be a function of the 

 temperature (t) } and if the finite differences (A log P) are known 

 in terms of the temperature, then the differentials (d . log P) are 

 similarly known ; and conversely. The existence of an exact 

 law being assumed, the simplest expression of that law will be 

 contained in the differential (d . log P) in terms of the tempera- 

 ture (/) . If such differential can be obtained from the observa- 

 tions of M. Regnault, and if such differential is of a simple form 

 and readily integrable, the law which governs the elastic forces 

 at all temperatures will be ascertained. 



The Table given by Regnault as the final result of his obser- 

 vations on the elastic forces of steam is found at page 624 of the 

 volume cited. In this Table the forces are stated in millimetres 

 of mercury for every degree Centigrade, from 32° C. below the 

 freezing-point of water to 230° C. above that point. At the 

 temperature 100° C, taken as the boiling-point of water, the 

 force is 760 millims., which is taken to represent the pressure of 

 one atmosphere. There is also given in the same Table a column 

 of differences or increments of forces for all consecutive intervals 

 of one degree of temperature. By means of this column the 

 value of the rate of increment of the force P, and consequently 

 the value of d . log P, may be easily obtained for any required 

 temperature. 



dV 

 The differential of hyperbolic logarithm P is -p-, which is the 



AP 



limit of p- when the intervals of temperature are indefinitely di- 

 minished. By taking AP equal to half the sum of the increment 

 preceding and the increment following the value of P for a given 

 temperature, and then dividing by P, we get the mean value of the 



rate of increment per degree f -p- or — +1 p ~* jfor the inter- 

 val of two degrees, one of which precedes and the other follows the 



