214 Prof. W. J. M. Rankine on the Thermal Energy 



impulse of small particles flying about in all directions within «i 

 closed vessel and rebounding from its sides. 



§ 5. Vortices with Heterotropic Action. — It is conceivable that, 

 in solid bodies, molecular vortices may be so arranged as to pro- 

 duce centrifugal pressures of different intensities in different 

 directions. In such cases it is to be recollected that the sum of 

 the mean values of cos 2 6 for the obliquities of any set of lines to 

 any three planes at right angles to each other is =1 ; whence it 

 follows that if//, p", and p"' be the mean intensities of the centri- 

 fugal pressures in any three orthogonal directions, we have 



p' + p»+p»' = p W <2 ; (4) 



that is to say, the sum of the mean intensities of the three centri- 

 fugal pressures in any three orthogonal directions is equal to twice 

 the energy of the steady circulation in a unit of volume. This 

 proposition was not in the paper of 1849-50, which was confined 

 to an isotropic arrangement of vortices. 



§ 6. Energy of the Periodical Disturbances.— -In the paper of 

 1849-50, p. L52, 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 a unit of volume the following 

 expression, 



pv* _ kpur 1 



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 circula- 

 tion, are to be deduced in each case from the results of experi- 

 ments on specific heat. 



Thus the energy of the disturbances in a unit of volume is 

 expressed by 



(h-l) p -f = \(k-\)p (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 



u 2 

 of the mass 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 



