170 Mr. C. Chree on Rotating Elastic 



in a circular section with ^ = '5, when the velocity allowed by 

 the stress-difference theory in a long cylinder is approximately 

 1*323 times that which it allows in a thin disk of the same 

 radius. For ordinary values of rj the differences are always 

 much less than this. Suppose for instance 97 = -25, then, 

 employing the suffixes 1 and 2 as before, and using O for the 

 limiting angular velocity in the thin disk, o> for that in the 

 long cylinder of the same cross section and material, we 

 deduce the following results : — 



Ratio of Limiting 



Table XIX. 



Velocities in Cylinder and Disk. 



b/a= 



•0. 



•2. 



•4. 



•6. 



•8. 1-0 



(o 2 /Q 2 = 



1 

 1-008 



1-019 

 1-007 



1-043 

 1-007 



1058 

 1-008 



1-078 1-104 

 1013 1-022 



§ 49. The solutions we have obtained both for thin disks 

 and long cylinders are, unless ^ = 0, only approximate, and the 

 principle of statically equivalent surface-forces on which they 

 are based is one as to whose degree of accuracy opinions may 

 differ. It is thus very desirable to subject our results to some 

 independent test. 



If we take the value of ( — 7) in (16), integrate it over a 

 cross section of the disk and divide the integral by irab, we 

 obtain the mean approach to the central plane of points 

 originally in a plane section at distance z. Representing this 

 mean value b}- ( — 8z) we easily find 



(-M/l) = V^(a* + b*) (106) 



Now for the change in length of a long elliptic cylinder of 

 the same section as the disk we find by (79) 



(-Sl/l)=^Xa*+V>) (107) 



But the right-hand sides of (106) and (107) are identical, or 

 we have 



(-Er/l) = (-Bl/l) = v »* P ii*E, . . . (108) 



