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



Substituting these values in the expressions for the strains 

 and stresses we find for all values of r and z 



9 = 0,\ 



du dw _ 

 dz dr ' 



.(22). 



and so rz = , 



Since rz is everywhere zero, the condition (6) over the edges is 

 exactly satisfied. The only surface condition left is (7), but this 

 we cannot exactly satisfy, unless m = n, by means of the present 

 solution. For, substituting the above values of the arbitrary con- 

 stants, we obtain from the expression for rr in terms of the strains 



™V=» = i (3m — n) A — 2na~ 2 D 



7m — n„ „ (m — n)(3m — n) „ „ /nox 



^ m 2 pa 2 - v — = — p r— L co 2 pz* (23), 



16m r 4>m(m + n) r 



rr r=a ' = similar expression, replacing a by a' (24). 



It is obvious we cannot make these stresses vanish for all values 

 of z. 



If the thickness 21 of the disk be of the same order of mag- 

 nitude as the radius a this failure renders the present method 

 inapplicable; but when l/a is small it is easy to obtain a solution 

 which according to Saint- Venant and other eminent authorities 

 must be very approximately exact except in the immediate neigh- 

 bourhood of the edges. 



The principle this solution is based on is that of statically 

 equivalent systems of loading. According to this principle when 

 a surface of an elastic solid has a small dimension — such as the 

 thickness of a thin disk — all systems of surface forces which in 

 their distribution along the small dimension are statically equi- 

 valent produce, except in the immediate neighbourhood of their 

 points of application, practically identical strains and stresses. 

 We may thus for practical purposes replace any system of surface 

 forces over the small dimension by any statically equivalent 

 system. 



For a discussion of this principle and illustrations of its 

 application, the reader is referred to Saint- Venant's Theorie de 

 L'Masticite...de Clebsch, p. 174 et seq. and p. 727 et seq., also to 

 Pearson's Elastical Researches of Barre de Saint-Venant, Arts. 8 

 and 9. 



In my previous treatment of this problem 1 , which was limited 

 to a complete disk, I determined the constant A so as to make 



1 Transactions, Vol. xiv. pp. 334 — 5, § 76, first two cases ; and Quarterly Journal, 

 Vol. xxiii. 1889, pp. 24—28. 



