of Molecular Vortices. 219 



a paper read to the British Association in 1865, and published 

 in the Philosophical Magazine for October of that year, a further 

 generalization is effected ; and it is shown that the general equa- 

 tion of thermodynamics follows from the supposition that sensible 

 heat consists 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 disturb- 

 ances 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 sec- 

 tion 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 cir- 

 culation. Hence the mean centrifugal tension, as this force may 

 be called, is proportional 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 tension is equiva- 

 lent, during a change of the figure and dimensions of all the 

 elementary circulating streams in a given body, may therefore 

 be expressed by multiplying the absolute temperature by the 

 change in the value of a function, to be afterwards determined, 

 of the dimensions, figure, and temperature. If to that function 

 be added a function which is the integral of the increment of 

 the energy of steady circulation divided by the absolute tempe- 

 rature, the sum is what I have elsewhere called the thermody- 

 namic function. Let it be denoted by (f>, and let dQ denote the 

 quantity of energy which must be communicated to the body 

 in order to produce the increment d . <j> in the thermodynamic 

 function at the mean absolute temperature f ; then we have 



dQ=rd.<f>; (18) 



and this, when the proper value has been assigned to the thermo- 

 dynamic function, is the general equation of thermodynamics. 

 The process of finding the value of the thermodynamic function 

 is well known, but a summary of it will be given here for the 

 sake of completeness : — 



