Solid Cylinders of Elliptic Section. 165 



must largely depend on how closely actual conditions are 

 represented by the assumed terminal conditions. This point 

 is not one which a mathematician is competent to decide, but 

 I do not see that either set given by Professor Grreenhill 

 seems more likely a priori than the combined set 



*=l=3=°- ■•■••• (") 



Thus, suppose the bearings on which the axle rests each of 

 breadth c, where c is, as must happen in practice, finite, 

 though small compared to the length of the cylinder. Further, 

 suppose the bearing to consist of a hollow circular cylinder, 

 inside which the circular axle can move freely without 

 appreciable friction. If the axle fitted the bearing exactly, 



then y and^ would each require to vanish both when x—0 



and when oa = c, with similar conditions of course at the other 

 end. Thus, without introducing any elastic principle, we 



du d u 

 should deduce that y, -f-, -~, and, perhaps, some higher 



differential coefficients must vanish at each end. We cannot, 



however, satisfy all these conditions with a solution such as 



(96) without having each of the 4 constants identically 



zero, in which case we reach no conclusion whatever as to 



instability. 



If we accept Professor Greenhill's elastic theory, the con- 



d 2 y 

 dition —^ — must hold over the terminal sections, unless 



their faces are in contact with some stops or held in some 

 way. Since the cylinder naturally shortens under rotation, 

 it is difficult to conceive how any system of support which 

 did not introduce very large frictional forces when the rota- 

 tion was slow could leave the terminal sections anything but 

 free when the rotation was rapid. Thus, if it be possible for 



dv 

 the supports to keep ~~ = at both ends, as Professor 



Greenhill supposes in his first set of terminal conditions, it 

 seems doubtful whether his theory leads to any conclusions 

 whatsoever as to instability. 



§ 45. It is only fair to recognize that the hypothesis that 

 the axle exactly fits the bearings — though apparently implied 

 in Professor GreenhnTs first set of terminal conditions — is 

 hardly possible in the strict mathematical sense, and some of 



