478 Prof. Osborne Reynolds on the Dilatancy of Media 



The general condition of the medium around a sphere which 

 has expanded in the medium is shown in Plate X. fig. 3, 

 which has been arrived at on the supposition that the sphere 

 is large compared with the grains. 



From a radius about 1 . 4R outwards the density gradually 

 increases, reaching a maximum density at infinity ; and at all 

 distances greater than 1 . 8R the law is expressed by 



de __ 1 



dr ~~ r n \ 



where n has some value greater than 3 depending on the 

 structure of the medium. 



Within the distance 1 . 4 R the variation is periodic, with a 

 rapidly diminishing period. In this condition, supposing the 

 medium of unlimited extent and the sphere smooth, the sphere 

 may move without causing further expansion, merely changing 

 the position of the distortion in the medium ; for the grains, 

 slipping over the sphere, would come back to their original 

 positions. It thus appears that smooth bodies would move 

 without resistance if the relation between the size of the grains 

 and bodies is such that the energy due to the relative motion 

 of the grains in immediate proximity may be neglected. 

 The kinetic energy of the motion of the medium would be 

 proportional to the volume of the ball multiplied by the density 

 of the medium and the square of the velocity. 



But the momentum might be infinite supposing the medium 

 infinite in extent, in which case a single sphere would be 

 held rigidly fixed. 



If we suppose two balls to expand instead of one, and sup- 

 pose the distortion of the medium for one ball to be the same 

 as if the other were not there, the result will be a compound 

 distortion. Since, however, the dilation does not bear a linear 

 relation to the distortion, the dilation resulting from the com- 

 pound distortion will not be the sum of the dilations for the 

 separate distortions unless we neglect the squares and products 

 of the distortions as small. 



Supposing the bodies so far apart that one or other of the 

 separate distortions caused at any point is small, then, retaining 

 squares and products, it appears that ihe resultant dilation at 

 any point will be less than the sum of the separate dilations 

 by quantities which are proportional to the products of the 

 separate distortions. 



The integrals of these terms through the space bounded by 

 spheres of radii R and L are expressed by finite terms, and 

 terms inversely proportional to L, which latter vanish if L is 



