NEWTON S PRINCIPIA. 299 



7 



CL CO 



and therefore the friction which is proportional both to the 

 velocity and the number of particles that rub against each 

 other will be proportional to 



2^^ a 



r * -r~ v T. 

 dr 



In order that the motion of any cylindrical element may 

 not change, the friction on its two sides must be equal. 

 Hence the expression 



r *d^ 



must be the same for all values of r ; call this value a ; 

 hence, integrating, we have 



* . -^* A ; : .i : -..,. 



where a and /3 are constants depending on the angular 

 velocities of the inner and outer cylinders. Supposing 

 these to be co l o&amp;gt; 2 , respectively, and r l r 2 to be the radii, we 

 have the two equations 



to find a and |3. 



In order that this state of motion may be &quot; stable,&quot; it 

 is necessary that the centrifugal force should increase 

 from the centre to the circumference, unless the fluid be 

 sufficiently tenacious to resist the change of motion that 

 might follow the inequality of pressure. 



