72 Mr. C. Chree on Rotating Elastic 



surface forces can differ appreciably from the theoretically 

 perfect solution, even when the surface forces are large, only 

 at distances from the surface which are comparable with the 

 small dimension. 



My solution in the Quarterly Journal contained a number 

 of arbitrary constants determined by the surface conditions. 

 The number of constants being insufficient to satisfy all the 

 surface conditions of the exact mathematical theory, the 

 conditions whose failure was selected in my paper as of least 

 importance were those which signified the vanishing of the 

 stress components parallel to the faces at every point of the 

 rim. While the other stresses vanished as required at every 

 point of the flat faces and the rim, these stress components on 

 the rim were taken to vanish only in the central plane £ = 0, 

 and elsewhere they were of the order z 2 of small quantities. 

 Professor Pearson's objection is that these stresses remained 

 uneqnilibrated , i. e. when treated by ordinary statics led to 

 a resultant force on every generator. The present solution is 

 free from this objection, and so presumably will be recognized 

 as a final solution for a thin disk by all who share Professor 

 Pearson's views. 



§ 2. As in my previous paper the origin is at the centre 

 of the disk, the axis of z being normal to the flat faces, 

 while the axes of x and y coincide with the major and minor 

 axes of the central section parallel to the faces. The peri- 

 meter of this section is the ellipse 



a-v+6-y=i, (i) 



whose major axis is 2a. 



The displacements are, as before, a, /3, 7, and the dilatation 

 A, where 



^+f+<k (2 ) 



ax ay dz 



For the stresses I shall employ the symmetrical notation of 

 Todhunter and Pearson's ' History/ so that the P, Q, R, S, T, U 



of my previous paper are replaced respectively by xx, yy-> 



zz, yz, zx, and xy. Also, instead of Thomson and Tait's 

 elastic constants m and n, I shall in general employ Young's 

 Modulus E, and Poisson's ratio* r], as more serviceable for 

 practical applications. 



The equations which ought to be satisfied when the right 

 cylinder (1), supposed of uniform density p and length 21, 

 rotates with uniform angular velocity co about its axis are nine 



* In what follows rj is assumed to lie between and *5 ; see Phil. 

 Mag. vol. xxxii. p. 236 (1891). 



