Endosmose and other Allied Phenomena. 61 



the resultant force and couple on the sphere equal to zero, it 

 will be sufficient to show that the force and couple-resultants 

 of the stress across a closed surface S x drawn in the fluid and 

 just enclosing the solid are zero. Using a common notation 

 for the components of stress at any point of the fluid we have 



P XX =~P + ^-^, &c.,&c.,"j 



**■ >-..... (25) 



where p is constant, by (12). The resultant stress parallel to 

 x across the complete boundary 2 of any space occupied by 

 fluid is 



§( l P X z + m P X y + n Pxz) d 2, 



where I, m, n are the direction-cosines of the normal to any 

 element d% of the boundary. This surface-integral is equal 

 to the volume-integral 



JjJ\ dx dy dz J 



dxdyd, 



taken throughout the interior of %, which vanishes, by (25), 

 since V 2 %=0. In a similar manner it may be shown that 

 the couple-resultant of the stress across X is zero. Now let 

 2 be made up of the surface Xi above defined, and of a sphere 

 S 3 of infinite radius, having its centre at the origin. It 

 follows that the stresses across 2i are statically equivalent to 

 those across 2 3 . And it easily follows from (22) that the 

 latter stresses are in equilibrium. 



It is remarkable that the velocity (24) is independent of 

 the size or shape of the particle, so long as its dimensions are 

 large in comparison with I. This velocity is, of course, to be 

 superposed on that of the fluid in the neighbourhood. For 

 instance, in the circumstances of Quincke's experiments we 

 have 



°-ttIP' 



and, therefore, for a suspended particle of the same nature as 

 the walls of the tube we should have for the absolute velocity 

 the value 



_3 aJ^ cE 

 2ttE 2 " |S 



when the particle is in the axis, and 



