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



length, a fair idea of this magnitude in any material may be 

 derived by supposing t] to vanish. But our solution is exact 

 when 77 = 0, and the maximum stress-difference and greatest 

 strain are then wholly independent of Ija. 



There is thus, on various grounds, a strong presumption that 

 in any isotropic material the limiting safe speeds allowed by a 

 complete elastic theory in short cylinders would not differ very 

 much from those which the elastic theory of the present paper 

 allows in long cylinders. Thus, even if Professor Greenhill's 

 method could be trusted when Ija ceases to be large, recourse 

 would require to be had to the strict elastic theory and not to 

 his formulae to fix a limiting speed. 



§ 30. In Professor Greenhill's paper, and in the discussion 

 which follows it, reference is made to propeller shafts in steamers. 

 These shafts, at least in large steamers, are of a length very great 

 compared to their diameter. The "Servia" for instance is said 

 to have a solid shaft 164 feet long and only 22^ inches in diameter. 

 It seems however the practice to support the shaft at intervals ; 

 in the "Servia" for example there are bearings making of the 

 shaft 8 lengths. In such a case the distance between the 

 bearings ought presumably to be regarded as the length 21 in 

 applying the instability formula. The value of Ija may perhaps 

 still be great enough for a legitimate application of the Greenhill 

 theory, but it is by no means so great that a consideration of 

 the magnitude of the elastic strains and stresses can be safely 

 dispensed with, at least in wrought-iron. When a shaft of this 

 kind is hollow, as is sometimes the case, the magnitudes of the 

 elastic strains and stresses given by our formulae should be looked 

 to, even if the material be the best steel. These remarks assume 

 of course that none but " centrifugal " forces act. In propeller 

 shafts under normal conditions this is far from true, there being 

 torsional and longitudinal forces as well, whose effects may be 

 large compared to those of the " centrifugal " forces. When 

 such a compound system of forces acts, the limiting speed ac- 

 cording to the elastic theory is to be got by superposing the 

 displacements arising from the several sources, and then ascribing 

 limiting values to the maximum stress- difference or greatest 

 strain given by the complete solution. It would however be 

 straying too far from our main subject to discuss this matter 

 further. 



§ 31. Reference has already been made to the comparatively 

 small difference between the limiting speeds allowed by the 

 elastic theories of " rupture " in long cylinders and in thin disks. 

 The exact relations between these speeds arc as follows: 



