1891.] Mr Chree, On thin rotating isotropic disks. 209 



For both the annular and the complete disks as and rz are by (22) 

 everywhere zero. The expressions for the strains and stresses in a 

 complete disk are correctly deduced from those in an annular 

 disk by leaving out all terms containing a! 2 . Allowing for the 

 change of notation, the solution for the displacements in a com- 

 plete disk differs from my previous one 1 only by terms in rff in 

 u, z¥ in w and I 2 in 8. Thus it only adds to the strains given by 

 the previous solution certain constant terms of order I 2 , and in no 

 respect modifies the conclusions derivable from that solution as 

 to the mode in which the strains and stresses alter with the 

 variables r, z. 



The expressions (32) and (33) for the stresses in an annular 

 disk when terms in I 2 and z 2 are neglected agree with those 

 which Professor Ewing 2 quotes as obtained by Grossmann 3 . 

 They likewise agree with those found by Clerk Maxwell 4 when 

 the error in the sign of his equation (59) pointed out by 

 Mr J. T. Nicolson 5 is corrected. The expression (31) for the 

 radial displacement when terms of order I 2 are neglected is iden- 

 tical with that given implicitly or explicitly by Maxwell, and by 

 Grossmann putting his N z = 0, and to the same degree of approxi- 

 mation (30) coincides with the value for the longitudinal displace- 

 ment to which Maxwell's theory would lead if fully worked. out. 

 I shall thus for brevity speak of the expressions our solution 

 supplies both for the complete and annular disks when terms of 

 order I 2 are neglected as constituting the Maxwell solution. 



The conclusion we are led to is that the methods of Maxwell 

 and Grossmann — which seem practically identical — while involving 

 inconsistencies 6 and certainly inconclusive from a strict theoretical 

 standpoint, perhaps even "paradoxical" as Professor Pearson 7 

 states, yet lead to results which if the present investigation can 

 be trusted are sufficiently exact for practical purposes so long as 

 the disk is very thin. 



From (33) it is obvious that w> is everywhere greater than 5r 

 in an annular disk. The same result follows from (38) for a com- 

 plete disk, except at the axis where the two stresses are equal. 



1 Quarterly Journal of Pure and Applied Mathematics, Vol. xxm. 1889, Equa- 

 tions (129), p. 28. 



2 Nature, 1891, p. 462. 



3 Verhandhmgen des Vereins zur Beforderung des Gcwerbfleisses, Berlin, 1883 

 pp. 216—226. 



4 Transactions of the Royal Society of Edinburgh, Vol. xx. Part i., 1853, pp. 

 Ill — 112; or Scientific Papers, Vol. i. p. 61. For corrections to Maxwell's second 

 equation (57) see Todhunter and Pearson's History of Elasticity, Vol. i. ft. -note 

 p. 827. 



5 Nature, 1891, p. 514. 



6 They lead to results inconsistent with one or both of the original assumptions, 

 viz. that rz is everywhere zero, and that zz if not also zero is independent of r and z 



7 Nature, 1891, p. 488. 



