286 Mr Chree, On Long Rotating Circular Cylinders. [Feb. 8, 



apparently solid should not have a cavity or cavities devoid of 

 molecules occupying the whole or a great portion of its axial 

 length. Would not then, it might be urged, such a solid cylinder 

 act according to our solution quite differently from another of the 

 same material in whose axial line there happened to be numerous 

 molecules ? The following seems a satisfactory explanation of this 

 difficulty : 



A hollow cylinder in the mathematical sense is one in which 

 7z and r? vanish over r = a' ; but this implies that a', even when 

 infinitely small compared to a, is still so great compared to mole- 

 cular distances, that the action of molecules separated by a 

 distance of order a' is inappreciable. There is thus no sudden 

 discontinuity, as our solution seems to imply, but a gradual trans- 

 ition as a increases from being a molecular distance to being a 

 distance so great that the mathematical conditions for a free 

 surface are satisfied. This transition stage is not within the 

 compass of the present mathematical theory ; but this is hardly a 

 matter of practical importance, for the mathematical conditions 

 are probably satisfied in any existing hollow cylinder as exactly 

 over its inner as over its outer surface. 



For brevity, a'ja = will be employed to denote the cylinder 

 that is hollow in the mathematical sense, but in which a /a is 

 extremely small. In the same way a /a = 1 will be employed to 

 denote a cylinder whose wall thickness a — a is extremely small 

 compared to a, though great compared to molecular distances. 



§ 6. Returning to our solution we see that £» vanishes when 

 7) = 0, so that all the surface conditions are then exactly satisfied. 

 The solution in this case is thus complete and applies to circular 

 cylinders of all shapes, to thin disks as well as to very long 

 cylinders. When r\ is small «» is very small compared to the 

 greatest stress w, and even when rj is \ the greatest value of «* 

 bears to the greatest value of JJ a ratio which is ^ in a solid 

 cylinder and not more than J^ in a hollow cylinder. When, how- 

 ever, 7) is | the greatest value of «£ in a solid cylinder is one half 

 the greatest value of JJ, and so may be by no means a small 

 stress. Now in the case of the thin disk the stresses which failed 

 to vanish over the edges were always very small compared to the 

 largest stresses. Thus, to all appearance, our solution for long 

 cylinders is not quite so satisfactory as that for thin disks unless 

 Poisson's ratio be small. 



§ 7. A striking difference between the effects of rotation on 

 thin disks and on long cylinders in which rj is not zero, is that 

 whereas in the former case originally plane sections perpendicular 

 to the axis of rotation become paraboloidal, in the latter case 



