37^ PROCEEDINGS OF THE AMERICAN ACADEMY § 7 



— P" ^= o, that is, if we express the two conditions always 

 assumed to hold in perfect gases, we shall have 



T 



as should be the case.* 



Substituting this value in the last equation, we find 

 jE =: P,' which is easily shown to hold for perfect gases. 



There is, therefore, every reason for confidence in the 

 formulae of this section, the applications of which will be 

 treated of when the ratio of/' to / has been determined. 



The next and several following sections will be devoted 

 to the relations which exist between the coeflEicients {£ and 

 e) and their derivatives, which, we shall find, can be estab- 

 lished independent of the value of/' -r- 1. It is sufficiently 

 obvious, from the preceding investigation, that some such 

 relations must exist; nevertheless these have not been 

 studied, and have been known only empirically hitherto. 



It is commonly supposed that in the experimental data 

 for the expansion of liquids, or those at least which have 

 been studied the most, little remains to be desired; such, 

 however, is not the case, as may easily be seen by com- 

 paring the results of different observers. There is prob- 

 ably no branch of physics in which the accuracy attainable 

 has been so far overestimated. Had the real uncertainty 

 of the determinations been recognized, the theoretical bear- 

 ings would undoubtedl}' have claimed more attention. 



§ 7. To investigate the laws of a liquid or solid, ex- 

 panding freely under heat, we put /^= o in equation III. 

 of the last section, which then becomes 



^ /— /' 



T 4/' -3/ 



I. 



• It may here, for convenience, be pointed out that c must not be confounded with the ab- 

 solute coefficient of expansion, being defined approximately as the ratio of increase of volume 

 to the volume itself for a rise of 1° in temperature The diff'erence, which is considerable for 

 gases, in the case of liquids and solids will be found to be slight. 



