35^ PROCEEDINGS OF THE AMERICAN ACADEMY § 2 



and that furthermore 



mv^ 



J^o mVo 



2 



whence combining, and denoting by vi, the mass of a 

 hydrogen molecule, and by vl the mean square of its 

 velocity at the freezing temperature of water ( 7^o= 273°), 

 we have, remembering that the density D is the quotient 

 of J/ and F, 



r VV^ -— _ , 1. 



3 ;// /; / — / 



the fundamental formula in the kinetic theory of liquids 

 and solids. If we suppose /' very small as compared with 

 /, this formula immediately reverts to one of the forms well 

 known in the theory of gases. 



Let us now suppose that a body on being heated 1° ex- 

 pands freely by the small ratio e; and that it is again com- 

 pressed at constant temperature to its original volume. 

 Designating by E the modulus of voluminal elasticity (after 

 Maxwell), or coefficient of resilience (after Everett) under 

 constant temperature, we shall require by definition an 

 external pressure, Ee^ equal and opposite to the increase of 

 internal pressure due to heat. The volume being un- 

 changed, the density must be the same; and if there be no 

 change in the molecular arrangement, the factor / -^ (/ — /') 

 must be unchanged; so that the only variable in equation I. 

 is T^ which has increased from a value, Z", to the value 

 T -\- 1°, thus causing an increase of the kinetic pressure 

 equal to 



3 ° m 7; /-/' T 



from which we see that 



3 m I^l — / 



