Mr. W. J. M. Rankine on the Mechanical Action of Heat. 7 



therij are the conditions which must be fulfilled, in order that a 

 group of vorticeSj of small size as compared with the bulk of an 

 atom, and of various diameters, may permanently coexist, whether 

 side by side, or end to end, in the atomic atmospheres of one 

 substance, or of various substances mixed. 



Fii'st, The mean elasticity must vary continuously ; which in- 

 volves the condition, that at the surface of contact of two vor- 

 tices of diflFerent substances, side by side, or end to end, the 

 respective densities at each point of contact must be inversely 

 proportional to the coefficients of elasticity. Hence the specific 

 gravities of the atmospheric parts of all substances, under pre- 

 cisely similar circumstances as to heat and molecular forces (a con- 

 dition realized in perfect gases at the same pressure and tempe- 

 rature), are inversely proportional to the coefficients of atmospheric 

 elasticity. Therefore let /x. represent the mass of the atmosphere 

 of one atom of any substance, b its coefficient of elasticity, and 

 n the number of atoms which, in the state of perfect gas, occupy 

 unity of volume under unity of pressure at the temperature of 

 melting ice ; — then 



nixh (I.) 



is a constant quantity for all substances. 



Secondly, The superficial elasticity of a vortex must not be a 

 function of its diameter : to fulfill which condition, the linear 

 velocity of revolution must be equal throughout all parts of each 

 individual vortex. 



Thirdly, In all contiguous vortices of the same substance, the 

 velocities of revolution must be equal ; and in contiguous vor- 

 tices of different substances, the squares of the velocities must be 

 proportional to the coefficients of elasticity of the molecular 

 atmospheres. 



The second and third conditions are those of equilibrium of 

 heat, and are equivalent to this law : — 



Tempekature is a function of the square of the velocity of 

 revolution in the molecular vortices divided by the coefficient of 

 elasticity of the atomic atmospheres ; or, 



Temperature = ^( -^ j, .... (II.) 



where tv represents that velocity. 



The mean elasticity which a vortex exerts endways is not 

 affected by its motion, being equal to 



l^P, (HI.) 



where p is its mean density. The superficial elasticity at its 

 lateral surfaces, however, is expressed by 



f +V (IV.) 



