1891.] Mr Chree, On thin rotating isotropic disks. 213 



The terms in zl 2 and z 3 in w would indicate that the longi- 

 tudinal compression is somewhat greater near the central plane 

 and somewhat less near the faces of the disk than according to 

 the Maxwell solution. 



For a first approximation confining our attention to the Max- 

 well solution in (31) and (36), we see that every point in the 

 disk, whether complete or annular, increases its distance from the 

 axis, and the transverse strain u/r is thus everywhere an extension. 



In the complete disk the radial strain is an extension inside 

 and a compression outside of a cylindrical surface co-axial with 

 the disk, and of radius r v given by 



ri = a^(S+ v )/(l+ v ) (44). 



This gives a value for r x less than a for all possible values of tj 

 except 0. Any annulus of the disk increases or diminishes in radial 

 thickness according as it lies inside or outside this surface. 



In the annular disk the increases da and da in the radii of 

 the edges, and d (a — a') in the radial thickness, a — a', are given 



by 



da = ( ^\(l- v )a 2 + (3+ v )a' 2 } (45), 



da = ^ {(3 + V ) a 2 + (1 - v ) a' 2 } (46), 



d ( Ct - a') = «> 2 P &-<*') {(a _ aJ _ v{a + a y } . . (47) . 



Thus the radial thickness is increased or diminished according as 



a/a< or > (I - ^77) 4- (1 + <f v ) (48). 



For the ratio of the radii in the annular disk whose radial thick- 

 ness is unaltered, we find 



r) = 0, a'ja = 1, 



7} = "25, a'ja = 3, 



7) = "36, a'ja = '25, 



7} = '5, a' '/a = 1716 approx. 



The radial strain given by the Maxwell solution in (31) is 



£K>-> m 



where f(r) = (1 - V )(S + v )(a* + a' 2 ) - 3 (1 - v 2 ) r 2 



- (l+7?)(3 + 77) ttVV 2 ... (50). 

 Since /(a) = - 2*, {(1 - v ) a 2 + (3 + v ) a"}, 



and /(«') = - 2 V {( 1 - 7 1 ) a' 2 + (3 + rj) d% 



17—2 



