516 Mr. Rankiue on the Centrifugal Theory of Elasticity, 



The coefficient a is determined by the following relation, ana- 

 logous to that of equation (2), between the densities 



whence 



Mm-'>-^- 



.Zbff 

 And the general value of the density is 



Mi-^'Kif (^) 



Making <2r=Z, and multiplying by the coefficient of elasticity 

 b, we obtain for the elasticity of the atmosphere, at the cylin- 

 drical surface of the vortex, 



bp, = bp+'0; (5<.) 



which exceeds the mean elasticity bp by a quantity equivalent 

 to the weight of a column of the mean density p, and of the 

 height due to the velocity w, and independent of the radius of 

 the vortex. 



Supposing a sphericallayer, therefore, to contain any number 

 of vortices of any diameter, in which the mean density is equal, 

 it is necessary to a permanent condition of that layer that the 

 velocities in all these vortices should be equal, in order that their 

 lateral elasticities may be equal. 



Although the mean elasticity at the plane end, or any plane 

 section at right angles to the axis of a vortex, is simply =bpj 

 being the same as if there were no motion, yet the elasticity is 

 variable from point to point, and the law of variation depends on 

 the velocity. Therefore if two vortices are placed end to end, it 

 is necessary to a stable condition of the fluid, not only that their 

 terminal planes should coincide, and that their mean elasticities 

 should be in equilibria, but also that their velocities should be 

 equal, or subject only to periodical deviations from a state of 

 equality. 



Therefore the mean velocity of vortical motion, independent 

 of small peiiodic variations, is the same throughout the whole 

 atomic atmosphere ; and the mean total velocity, independent of 

 small periodic variations, being uniformly distributed also, the 

 7ns viva of the fonner may be expressed as a constant fraction of 

 that of the latter, so that 



'o'=j, (5*) 



