112 On a Modified Theory of Gravitation. 



perfectly incompressible liquid, through which vortices are 

 distributed, constitutes a kinetic model of a slightly com- 

 pressible medium such as we have assumed the aether to be, 

 provided some or all of the vortices are coreless. One way 

 in which a very minute degree of compressibility may be 

 represented, is by supposing only a small proportion of the 

 vortices present to be coreless. When such a medium is 

 subjected to increased pressure, so as to diminish the volume 

 occupied by the vacuous cores, the circulation around each 

 vortex remaining unaltered, the energy of the turbulence will 

 be increased by an amount equal to the work done by the 

 pressure, and this additional energy is to be regarded as 

 potential when the turbulently moving liquid is treated as a 

 continuous medium. 



74. If we suppose each core, whether vacuous or con- 

 sisting of rotationally moving liquid, to be of very small 

 diameter in relation to the radius of curvature of its " curved 

 axis," and very small also in comparison with the distance 

 between neighbouring vortices, an expression may readily be 

 obtained for the compressional elasticity of the medium. If 

 ■p is the mean pressure, we shall have, for any point in the 

 liquid sufficiently remote from vortices, 



P SttV ~ 8ttV ' " " " ■ ' ^* J 



where r 1} r 2 , . . . are the radii of the various vacuous cores, 

 Hi, I2 2 , . . . the circulations around them, and p the density 

 of the liquid. Hence, extending the summation to the unit 

 of volume, and calling l x , l 2 , . . . the lengths of the respective 

 coreless vortices comprised in that volume, 



eg|!= 3Bfar» - U, say; .... (100) 



so that U is the total volume of vacuous cores comprised in 

 unit volume of the turbulent liquid. Also, assuming the I's 

 to be invariable, 



-f -4^-f *£' • • • « 



where, corresponding with any one vortex of strength H, 

 there is a volume v of vacuous core comprised within the 

 particular unit volume now considered. The left-hand 

 member of (101) represents evidently the reciprocal of the 

 bulk-modulus of elasticity appertaining to the medium as 



