210 Mr Chree, On thin rotating isotropic disks. [May 4, 



Also zz and rz vanish at every point, thus both in the annular 

 and in the complete disk u is everywhere the stress-difference 

 and u/r the greatest strain. Both quantities for any given value 

 of r are greatest when z = 0, and for any given value of z are 

 greatest when r = for the complete disk, or r = a for the 

 annular. They are thus according to the solution greatest in 

 the central plane, at the centre of a complete disk and at the 

 inner edge of an annular. 



According, however, to the principle of statically equivalent 

 surface forces our solution does not strictly apply for values of r 

 which differ from a or from a' by quantities which do not exceed 

 several multiples of I. In other words, it possibly may give values 

 for the strains and stresses over the edges differing from the true 

 values by terms of the order f. Thus in determining the greatest 

 values of the stress -difference or greatest strain, which occur 

 at or immediately adjacent to the inner edge in an annular 

 disk, we are not warranted in retaining terms of this order of 

 small quantities. I thus propose in determining these quantities 

 to neglect terms of this order as being of doubtful accuracy, at 

 least in an annular disk, and of insignificant magnitude in any 

 thin disk. It should be noticed, however, that our complete 

 solution gives at all radial distances larger values for the strains 

 and stresses in the central plane than when terms in I 2 are neg- 

 lected. Thus it would certainly only be prudent to regard the 

 values we are about to find from the Maxwell solution for the 

 maximum stress-difference and greatest strain as minima, which 

 in all probability are exceeded in any actual case. In a thin 

 disk, however, the true values can exceed these only by small 

 terms, of order (//a) 2 at least. 



Neglecting then terms in I 2 and z\ we find for the maximum 

 stress-difference 8 and the largest value of the greatest strain s — 



In a complete disk, ) * s _ 



F \Es, = (1- V )S 1 (40), 



In an annular disk, 8 2 = E\ = \a> 2 P [a 2 (3 + y) + a 2 (1 - y)} . . .(41). 



Since 8 2 = E\ the maximum stress-difference and greatest strain 

 theories lead to identically the same result for the so-called 

 "tendency to rupture" — i.e. approach to limit of linear elasticity 

 — in the annular disk. In the complete disk the maximum stress- 

 difference theory assigns for all possible values of y, except 0, 

 a lower limit than the other for the safe velocity of rotation. 



Supposing g) 5 and w 2 the limiting safe angular velocities in a 

 complete and in an annular disk of the same material and external 

 radius, then putting in succession 8 % = 8 2 and s t =s ? , we find 



