698 Lord Rayleigh on the Motion of 



§ 2. In attempting to go further, one of the first questions 

 to suggest itself is whether similar conclusions are applicable 

 to bodies of other forms. The consideration of this subject 

 is often facilitated by use of the well-known analogy between 

 the motion of a viscous fluid, when the square of the motion 

 is neglected, and the displacements of an elastic solid. 

 Suppose that in the latter case the solid is bounded by two 

 closed surfaces, one of which completely envelopes the other. 

 Whatever displacements {a, 7, ft) be imposed at these two 

 surfaces, there must be a corresponding configuration of 

 equilibrium, satisfying certain differential equations. If the 

 solid be incompressible, the otherwise arbitrary boundary 

 displacements must be chosen subject to this condition. The 

 same conclusion applies in two dimensions, where the bounding 

 surfaces reduce to cylinders with parallel generating lines. 

 For our present purpose we may suppose that at the outer 

 surface the displacements are zero. 



The contrast between the three-dimensional and two- 

 dimensional cases arises when the outer surface is made to 

 pass off to infinity. In the former case, where the inner 

 surface is supposed to be limited in all directions, the dis- 

 placements there imposed diminish, on receding from it, in 

 such a manner that when the outer surface is removed to a 

 sufficient distance no further sensible change occurs. In the 

 two-dimensional case the inner surface extends to infinity, 

 and the displacement affects sensibly points however distant, 

 provided the outer surface be still further and sufficiently 

 removed. 



The nature of the distinction may be illustrated by a 

 simple example relating to the conduction of heat through 

 a uniform medium. If the temperature v be unity on the 

 surface of the sphere r=.a, and vanish when r = b, the steady 

 state is expressed by 





^C- 1 ) <*> 



When b is made infinite, v assumes the limiting form a jr. 

 In the corresponding problem for coaxal cylinders of radii a 

 and b we have 



log 5 - log r ..... (3) 



log b— log a 



But here there is no limiting form when b is made infinite. 

 However great r may be, v is small when b exceeds r by 



