166 Mr. C. Chree on Rotating Elastic 



the above difficulties might be avoided by a judicious use of 

 this fact. Thus, if we suppose the radius of the bearing to 

 exceed that of the axle by or*, we could satisfy the set (98) of 

 terminal conditions by putting in (96) 



A=B = D = 0, 2fil = 7r, .... (100) 

 whence 



y = sin (war/2/) (101) 



The constant C is not, however, altogether arbitrary. The 

 axle must press against the bearing at the section x=c, and 

 it seems most reasonable to suppose it would also press against 

 it at the section # = 0, though this is not apparently absolutely 

 necessary. On the first view we must have 



C = 4Zo>/(tt<0 ; 



on the second view C is only limited to being less than this 

 value. This solution, it will be noticed, answers to one definite 

 value of //,, f . e. to one given angular velocity. With either a 

 smaller or a greater angular velocity — with definite exceptions 

 in the latter case to be noticed presently — the solution (96) can 

 satisfy the terminal conditions only when C vanishes as 

 well as the other three arbitrary constants. It would thus 

 appear that the meaning of our solution is as follows : — 

 Supposing the angular velocity gradually to increase from 

 zero, the axis of the cylinder must remain straight until the 

 velocity is reached for which 2/z/ = 7r. The axis may remain 

 straight when the velocity passes through this value, or it may 

 not. If it remains straight for this critical value it must con- 

 tinue straight while the velocity continues to increase until 

 the value is reached for which 2jjlI = 27t, when again it is possible 

 for it to bend. It may, however, happen that as the velocity 

 is attained for which 2/x/ = 7r the axis bends. The bending 

 will take place suddenly, the consequence being an impulse 

 between the cylinder and the sections x=c, x=2l — c of the 

 bearings. This may suffice of course to smash the bearings 

 or the cylinder, in which case the instability theory may per- 

 haps be considered satisfactory. If, however, this impact 

 does not smash either the bearings or the cylinder, the danger 

 seems to be passed unless the rotation be kept exactly at the 

 critical velocity. Now it is clear that if Ihr/c be very small, 

 the resultant " centrifugal " force answering to the displace- 

 ment (101) is small, and the impulse on the bearings may be 

 but trifling. Thus the danger which the instability theory 



* This implies, however, that when the axis of the cylinder bends, the 

 line joining its ends ceases to be fixed in space. 



