164 Mr. C. Chree on Rotating Elastic 



the action of the " centrifugal " forces the same as if the mass 

 were collected in the axis. This axial distribution of force is 

 then supposed in equilibrium with the elastic forces, the dis- 

 tribution of stress over each cross section being assumed to 

 give a couple as in the ordinary Bernoulli-Eulerian treatment 

 of beams under flexure. This leads to a differential equation 

 of the form 



S-<*-fc w 



where x is measured along the line joining the ends of the 

 cylinder's axis, y is the distance from this line of a point in 

 the axis in its displaced position, while fi is a constant depend- 

 ing on the velocity, material, and dimensions of the cylinder. 

 Supposing the origin at an end of the axis, we may represent 

 the solution of (95) by 



y = A sinh fix -f B cosh fix + C sin fix + D cos fix, . (96) 



where A, B, C, D are constants depending on the terminal 

 conditions. Prof. Greenhill takes two alternative sets of 

 conditions : — 



(1) y = 0=^-, at both ends, . . . .(97) 



(2) y=0=g, „ „ .... (98) 



These lead to different results for the limiting speed. 



§ 44. The condition y = merely fixes the origin of co- 

 ordinates, assuming the line joining the ends of the axis to be 



fixed in space. The condition jf =0 at an end signifies that 



the direction of the axis is there fixed, while -^ —0 signifies 



the vanishing of the elastic couple given by the Bernoulli- 

 Eulerian method of treatment*. This latter condition is thus 

 required by Professor Greenhill' s theory when no applied 

 couple acts over a terminal section. Now, supposing Professor 

 Greenhill's method of reducing the physical problem to a 

 mathematical form sufficiently exact, so that (95) is satis- 

 factory, it is clear that the reliance to be placed on his results 



cPy 

 * Prof. Greenhill introduces the condition -~=0 on his p. 200, with- 

 out explicit reference to his elastic theory, but the above is, I believe, the 

 explanation he had in view. 



