560 PROFESSOR RANKINE ON THE THERMAL ENERGY 



that is to say, the sum of the mean intensities of the three centrifugal pressures 

 in any three orthogonal directions is equal to twice the energy of the steady circula- 

 tion in an unit of volume. This proposition was not in the paper of 1 849-50, 

 which was confined to an isotropic arrangement of vortices. 



§ 6. Energy of the Periodical Disturbances. — In the paper of 1849-50, p. 152, 

 equation x., the energy of the periodical disturbances was taken into account by 

 multiplying the energy of the steady circulation by a factor k greater than unity; 

 thus giving for the total energy in an unit of volume the following expression, 



pv 2, kpvj 2 

 T = ~2~ ' 



in which v 2 denotes the mean of the squares of the resultant velocities of the 

 particles with their combined motions. The values of the factor k, being the 

 ratio which the total energy of the molecular motions bears to the energy of the 

 steady circulation, are to be deduced in each case from the results of experiments 

 on specific heat. 



Thus the energy of the disturbances in an unit of volume is expressed by 



(*-!)'£= S(*-l> .... (5). 



It may now be observed, in addition, that the energy of the disturbances may, 

 and indeed must, be at times partly potential as well as actual ; in other words, 

 partly due to displacement as well as to fluctuation of velocity. 



Let ± u be the greatest fluctuation of velocity; then a particle of the mass 



u 2 



unity has the energy -~ due to that fluctuation, in addition to the energy due 



to the steady circulation. It is only at the instants of greatest disturbance of 

 velocity that the energy is all actual : at every other instant the energy is partly 

 potential. Hence v 2 = km 2 may be taken to denote, not the square of an actual 

 velocity common to all the particles, but the value to which the square of the 

 velocity of the particles would rise, if all the energy of the disturbances, actual 

 and potential, were expended in increasing the velocity of steady circulation. 



§ 7. Total Energy of Thermal Motions. — The total energy of the motion, com- 

 pounded of steady circulation and periodical disturbances, in an unit of volume, 

 is expressed, as in the paper of 1849-50, by the following equation, which also 

 shows its relation to the centrifugal pressure, 



~2~ = ^p • • ( 6 ); 



in which (to recapitulate the notation) p is the mean density ; to the velocity of 

 steady circulation ; the centrifugal pressure p is expressed in absolute units of 



