[56] PROFESSOR STOKES, ON THE EFFECT OF THE INTERNAL FRICTION 



The correctness of this explanation is confirmed by the following considerations. Suppose 

 that F (r) had been given by the equation 



F (r) = Ar- 1 + Br + CV" 5 + Dr s 



instead of (130), 5 being a small positive quantity. On this supposition it would have been 

 possible to satisfy all the equations of condition, and therefore steady motion would have been 

 possible. By determining the arbitrary constants, and substituting in ■%, we should have 

 obtained 



y\, = aV{ s + . - Uin 0, 



Since $ is supposed to be extremely small, it follows from these expressions that when r is not 

 greater than a moderate multiple of a, the velocities R, Q are extremely small ; but, however 

 small be 8, we have only to go far enough from the cylinder in order to find velocities as 

 nearly equal to - VcosO, + Fsin Q as we please. But the distance from the cylinder to which 

 we must proceed in order to find velocities R, 9 which do not differ from their limiting values 

 — Fcos#, + Fsin 9 by more than certain given quantities, increases indefinitely as $ decreases. 

 Hence, restoring to the fluid and the cylinder the velocity V, we see that in the neighbourhood 

 of the cylinder the motion of the fluid does not sensibly differ from a motion of translation, the 

 same as that of the cylinder itself, while the distance to which the cylinder exerts a sensible 

 influence in disturbing the motion of the fluid increases indefinitely as $ decreases. 



48. When we have formed the equations of motion of a fluid on any particular dynamical 

 hypothesis, it becomes a perfectly definite mathematical problem to determine the motion of the 

 fluid when a given solid, initially at rest as well as the fluid, is moved in a given manner, or 

 to discuss the character of the analytical solution in any extreme case proposed. It is quite 

 another thing to enquire how far the principles which furnished the mathematical data of the 

 problem may be modified in extreme cases, or what will be the nature of the actual motion in 

 such cases. Let us regard in this point of view the case considered in the preceding article as 

 a mathematical problem. When the quantity of fluid carried with the cylinder becomes con- 

 siderable compared with the quantity displaced, it would seem that the motion must become 

 unstable, in the sense in which the motion of a sphere rolling down the highest generating line 

 of an inclined cylinder may. be said to be unstable. But besides the instability, it may not be 

 safe in such an extreme case to neglect the terms depending on the square of the velocity, not 

 that they become unusually large in themselves, but only unusually large compared with 

 the terms retained, because when the relative motions of neighbouring portions of the fluid 

 become very small, the tangential pressures which arise from friction become very small like- 

 wise. 



