212 Mr Ghree, On thin rotating isotropic disks. [May 4, 



an originally plane section at distance z from the central plane is 

 in both the complete and annular disks 



2E+{>n(i + V )a>*pz}. 



It thus depends merely on the angular velocity, the nature of the 

 material and the original distance from the central plane. Its 

 reciprocal, and so the curvature at the vertex of the paraboloid, 

 varies directly as the square of the angular velocity and as the 

 distance from the central plane. 



The amount by which the axial point on an originally plane 

 section parallel to the faces — of course an imaginary point in an 

 annular disk — approaches the central plane varies as the square 

 of the radius of the complete disk. The corresponding approach 

 in an annular disk varies as the sum of the squares of the radii 

 of its edges, and is greater than in a complete disk of the 

 same external radius. - The magnitudes of the reductions in the 

 thickness, 21, of the disks will be seen from the following data — 



a,t inner edge ^ (o 2 plr) {(3 + rj) a 2 + (1 — rj) a' 2 }, 

 In annular disk -| 



at outer edge ^ a? ply {(3 + rj) a' 2 + (1 — rj) a 2 }, 



In complete disk 



at axis ^=, a> 2 pln (3 + rj) a 2 , 



at outer edge jr-^, o> 2 plr) (1 — rj) a 2 



The terms in zl 2 and z 3 in (30) and (35) cut out when z = ± I, 

 so that as regards the preceding results as to the change of thick- 

 ness there is an exact agreement between the Maxwell solution 

 and the more complete solution. 



It will be noticed that the reduction in thickness at the inner 

 edge of an annular disk equals the reduction at the axis of a 

 complete disk equal in radius to its outer edge, together with the 

 reduction at the rim of a complete disk equal in radius to its 

 inner edge, the thicknesses, materials and angular velocities being 

 the same in each case. Also the reduction in thickness at the 

 outer edge of an annular disk exceeds what it would be if the 

 disk were complete by the reduction at the axis of a complete 

 disk equal in radius to its inner edge. 



The longitudinal strain dw/dz parallel to the axis is a com- 

 pression at every point, unless rj = 0, both in the complete and 

 annular disks, and diminishes numerically as r increases. For 

 ordinary materials it is a quantity of the same order of magnitude 

 as the radial and transverse strains, but it vanishes throughout if 



7? = 0. 



