1891.] Mr Ghree, On thin rotating isotropic disks. 203 



We may suppose the disk at rest, acted on by a " centrifugal 

 force" (o 2 pr per unit of volume. Thus the internal equations are 

 the following two 1 



drr drz . rr — <p<p . 2 a /n\ 



-7- + -7-H — +et>pr = (2), 



dr dz r ' 



dp + 7; + d£ =0 



dr r dz v ' 



Let 21 denote the thickness of the disk, a the radius of its outer 

 a' that of its inner cylindrical surface, or edge. Then supposing 

 the disk exposed to no surface forces, the solution ought to satisfy 

 the following surface conditions 2 — 



fz = (4), 



over the flat faces z = + I, i~ _ /KX 



" l« = (5), 



over the edges r — a and r=a, J 



[rr = (7). 



Substituting for the stresses their expressions in terms of the 

 strains and using (1), we easily transform (2) and (3) respectively 

 into 



, N d8 d { (du dw\) „ „ 



(»+.), - + „ s | r ^ _ _j| =_ kV (8) , 



f . d8 d [ (du dw\) 



(m+n)r &-"di-\ r U-^rr (9) - 



Differentiating (8) with respect to r and (9) with respect to z, 

 then adding and dividing out by (ra -t- n) r, we get 



dr 2 r dr dz 2 m + n 



Of this a particular solution is 



S = -o) 2 pr 2 ^2{m + )i). 



A complementary solution in ascending powers of r and z with 

 arbitrary constants can be obtained, as I have shown in a previous 

 paper 3 . Of this we require for our present purpose only the 

 constant term and that of the second degree, or 



8 = A + C(z 2 -±r 2 ). 



1 Pearson's Elastical Researches of Barre de Saint-Venant, foot-note, p. 79, or 

 Ibbetson's Mathematical Theory of Perfectly Elastic Solids, p. 239. 



2 Ibbetson's Mathematical Theory 1. c. , or Todhunter and Pearson's History 



of Elasticity, Vol. i. Art. 614. 



3 Transactions, Vol. xiv. p. 328 et seq. 



