564 



PROFESSOR RANKINE ON THE THERMAL ENERGY 



The following are some examples of the results of calculations by for- 

 mulae (15) and (17) : — 



Substance, 



Atmospheric air, 

 Nitrogen, 

 Oxygen, . 

 Hydrogen, 

 Steam-gas, 



& 



c 



k 



&*-»> 



1-408 



1-634 



0-951 



1-409 



1-630 



0-945 



1-400 



1067 



1000 



1-413 



1-614 



0-921 



1-297 



2242 



1-863 



§11. General Equation of 'Thermodynamics — In the paper of 1849-50, pp. 

 158 to 164, the general equation of thermodynamics (equation 6 of that paper' 

 p. 161) is deduced from the hypothesis of molecular vortices, on the supposition 

 of a special form and arrangement of the vortices. In a subsequent paper, " On 

 the Centrifugal Theory of Elasticity," read to the Royal Society of Edinburgh in 

 December 1851 (" Transactions," vol. xx. pp. 433 to 436), the same general 

 equation (being equation 25 of the latter paper, p. 436) is deduced from the 

 hypothesis of molecular vortices, without any special supposition as to the form 

 and arrangement of the vortices, but with certain assumptions as to the laws of 

 the elasticity of the matter which moves in them. In a paper read to the British 

 Association in 1865, and published in the "Philosophical Magazine" for October 

 of that year, a further generalisation is effected ; and it is shown that the general 

 equation of thermodynamics follows from the supposition, that sensible heat con- 

 sists in any kind of steady molecular motion within limited spaces, without any 

 assumption either as to the figures of vortices, or as to the special properties of 

 the matter that moves in them. The object of this section of the present paper 

 is to show how the same general equation is deducible from the hypothesis of 

 molecular vortices, as stated at the commencement of the paper ; that is, freed 

 from all special suppositions except that of a steady circulation, combined with 

 periodical disturbances of speed, whose energy may bear any proportion, constant 

 or variable, to that of the steady circulation. 



The forces by which an elementary circulating stream, whether flowing with 

 a steady or with a fluctuating speed, is kept in a given state of motion, and of a 

 definite figure and dimensions, are equivalent in their action to a tension exerted 

 at each cross-section of the stream, of an amount which, at a given cross-section, 

 and at a given instant, is expressed in absolute units of force by the product of 

 the mass which flows along the stream in a second into the velocity of flow at 

 that cross-section and instant. The mean value of the tension is the product of 

 the same mass into the mean velocity ; that is, into the velocity of steady circu- 

 lation. Hence the mean centrifugal tension, as this force may be called, is pro- 

 portional to the square of the velocity of steady circulation, and therefore to the 

 absolute temperature; and the work done by the forces to which the virtual 



