84 Mr. C. Chree on Rotating Elastic 



of 7] and b/a. Also let 1/Rj and 1/R 2 be the values of 1/R in 

 (38)— i. e. of ^R^ + Rg -1 ) in (37)— which answer to the 

 limiting speeds co^ and o) 2 a respectively. Then it is easy to 

 prove : 



«^=(B/E)(J/«)i0(l+?){AO»» */«)}*> • • • • (39) 

 a/B,=(n/aHl+v)if/fh _*/«)}' (40) 



In a disk of given shape and material the limiting curvature 

 according to either formula varies as l/a 2 . As a numerical 

 example, suppose the disk circular and of a material for which 

 7] = *25 ; then by Tables I. and II., 



(a/50-r (S/E) = (l/a) x -25 x 1*25 x (1-569) 2 , 



= (l/a) x *769 approximately, 



(a/R 2 )H-5 =(Z/a)xr026 „ 



With good steel a strain 5 = '001 would answer to a stress 

 of some 12 or 13 tons weight per square inch, a load which is 

 apparently nearly reached in some engineering structures. 

 If we suppose the stress-strain relations linear up to this 

 point, we could by rotation produce at the centre of a face of 

 a disk in which a/ 1 =10 a curvature bearing to that of the 

 rim of the disk the ratio 1 : 10 4 . 



§17. The previous data as to the curvature produced by 

 rotation in planes parallel to the faces also enable us to form 

 an idea of the character of the other displacements. As 



shown by (12) the stresses yz and,^ are everywhere zero, i. e. 

 at every point of the disk 



f + <t = 0=¥+p (41) 



dz dy dx dz v ' 



This signifies that all material lines which before strain 

 were perpendicular to the axis of z continue during rotation to 

 cut orthogonally the material lines which before strain were 

 parallel to the axis of z. The latter set of material lines thus 

 become during rotation orthogonal to surfaces which accord- 

 ing to the first approximation are elliptic paraboloids whose 

 axis is the axis of rotation. They thus become concave to 

 the axis of rotation, and their curvature increases with their 

 distance from that axis. At a given distance from the axis 

 of rotation this curvature is most conspicuous in the plane of 

 yz containing the minor axis of the disk. 



§ 18. For the limiting value of rj, for which our solution 

 is complete, the phenomena are much simpler. The faces of 



