1891.] Mr Chree, On thin rotating isotropic disks. 207 



rr r=a vanish for a given value of z, viz. either z = or z = ± I. In 

 either case we are left with a system of unequilibrated normal 

 forces along each generator of the edge. On this ground Professor 

 Pearson has recently 1 expressed his opinion that my solution 

 cannot be regarded as " final " even for a thin disk. As the 

 normal forces in question are of the order of the square of the 

 thickness of the disk, I am not altogether sure what weight may 

 be attached to this criticism. In deference however to Professor 

 Pearson's opinion, and to what I believe the view Saint-Venant 

 would have taken, I propose the following method of solution 

 which removes at least this objection. 



It consists in determining A and D from the equations 

 •+i 



ft r=a dz =0 (25), 



-+i 



£ r=a >dz = (26). 



■i 



This still leaves normal stresses of the order of the square of the 

 thickness over the edges, but the forces along each generator of an 

 edge form a system in statical equilibrium. Thus according to 

 the principle of statically equivalent systems, the solution we 

 shall obtain — which must be strictly limited to thin disks — gives 

 expressions for the strains and stresses which can differ sensibly 

 from those supplied by the complete solution only in the im- 

 mediate neighbourhood of the edges. 



For the case of a complete disk D must vanish and A is to be 

 determined by (25). 



For the annular disk we find from (25) and (26) 

 . 7 m — n 2/2 «\ . m — n „ 7 A 



&m(3m — n) r bm(m + n) r 



T\ ' "'' ^ 2 2/2 



JJ = -7^ (o pa a 



Szmn 



.(27). 



Also from equations (21) we have G and £ determined explicitly, 

 and a found in terms of A. Thus all the constants of our solution 

 are determined. For a complete disk we have only to put D = 0, 

 and a' = in the expression for A in (27). 



The physical results attainable from the solution will perhaps 

 be rendered more practically serviceable by replacing the m, n of 

 our previous work by Young's modulus E and Poisson's ratio v. 

 To express the values obtained above for the arbitrary constants 

 in terms of E and n we require the relations 



m = %E/{(l-2 v )(l+ v )} r 



1 Nature, 1891, p. 488. 



