98 Mr. C. Chree on Rotating Elastic 



Now the present and the previous solutions assign precisely 

 the same values for the curvature of lines originally parallel 

 to z throughout the whole disk. Thus unless the fact that 



xx 2 and yy 2 have a term variable with z, whereas xx$ and 



yy % have not, constitutes a fundamental difference, we should 

 expect the curvature supplied by the present solution to be 

 everywhere exact to the present degree of approximation. 



This seems almost conclusive so far as concerns points not 

 in the immediate neighbourhood of the rim. For if there be 

 any truth in the theory of statically equivalent load systems, 



the influence of a stress system xx 2 , yy 2j whose statical 

 resultant over a generator is zero, must be small except in the 

 immediate neighbourhood of its application compared to the 



influence of a system xx 3 , yy 3 , where the stresses at every 

 point of a generator have the same sign, while their order of 

 magnitude is the same as that of the other system. 



Near the rim the existence of the terms in z 2 in xx 2 , yy 2 

 renders it probable that this stress system would have more 



effect on the curvature than the system xx s , yy 3 . Still in the 

 latter case, though the stress is uniform along a generator, 

 its effect at a neighbouring point in the disk must be largely 

 influenced by the greater or less proximity of that point to a 

 face of the disk. Thus, while one might reasonably look for 

 a greater curvature in the first case than in the second, one 

 would hardly expect the curvatures in the two cases to be of 

 different orders of small quantities. 



These considerations point strongly to the conclusion that 

 the expressions we are about to find for the curvature of lines 

 originally parallel to the axis of rotation are exact to the 

 present degree of approximation for all points not very close 

 to the rim, and that even at the rim itself the error is likely 

 to be small. General reasoning of this kind is, however, at 

 best unsatisfactory, and I do not regard the results for the 

 curvature, at least close to the rim, as deserving the same 

 confidence as the other results of this paper. 



§ 31. In determining the curvature of a line through the 

 point (x, y, 0) originally parallel to the axis of rotation, we 

 require to know the osculating plane of the curve it forms. 

 Now the curve, as we have seen, is orthogonal to the para- 

 boloids (29)— strictly so at the faces of ihe disk, sufficiently 

 nearly so elsewhere—; thus the osculating plane at {x, y, 0) 

 contains the normal to the ellipse of equal longitudinal dis- 

 placement through that point. Thus the principal radius of 



