OF FLUIDS ON THE MOTION OF PENDULUMS. [99] 



Note A, Article 65. 



Let us apply the general equations (2), (3) to the fluid surrounding a solid of revolution 

 which turns about its axis, with either a uniform or a variable motion, supposing the fluid to 

 have been initially either at rest, or moving in annuli about the axis of symmetry. 



In the first place we may observe, that the fluid will always move in annuli about the axis 

 of symmetry. For let P be any point of space, and L any line passing through P, and 

 lying in a plane drawn through P and through the axis of symmetry ; and at the end of the 

 time t let u be the velocity at P resolved along L. Now consider a second case of motion, 

 differing from the first in having the angular velocity of the solid and the initial velocity of 

 the fluid reversed, every thing else being the same as before. It follows from symmetry, that 

 at the end of the time t the velocity at P resolved along L will be equal to u, since the 

 motion of the solid and the initial motion of the fluid, which form the data of the one problem, 

 differ from the corresponding quantities in the other problem only as regards the distinction 

 between one way round and the other way round, which has no relation to the distinction 

 between to and fro in the direction of a line lying in a plane passing through the axis of 

 rotation. But since all our equations are linear as regards the velocity, it follows that in the 

 second problem the velocity will be the same as in the first, with a contrary sign, and therefore 

 the velocity at P in the direction of the line L will be equal to - u . Hence u = — u, and 

 therefore u = 0, and therefore the whole motion takes place in annuli about the axis of 

 rotation. 



Let the axis of rotation be taken for the axis of z ; let w be the angle which a plane 

 passing through this axis and through the point P makes with the plane of xy, and let v be 

 the velocity at P. Then 



u = - v sin a), v = v cos w, w = 0, 



and all the unknown quantities of the problem are functions of t, z, and ar, where 

 ■z«r = \/ (tf 2 + y s ). Substituting in equations (2) the above values of w, v, and w, and after 

 differentiation putting u> = 0, as we are at liberty to do, we get 



dor dz 



'drv <Pv 1 dv v \ dv' 



[d'v d-v 1 dv v\ dv 



Md? + ^ + - d^-^) = P~dT (179) 



The first two of these equations give p = a constant, or rather p = a function of t, which for 

 the same reason as in Art. 7 we have a right to suppose to be equal to zero. The third 

 equation combined with the equations of condition serves to determine v. 



Now in the particular case of an oscillating disk, the equation (179) becomes according to 

 the mode of approximation adopted in Art. 8 



dV dv 



fi d^ = PTt' (180 > 



37—2 



