Solid Cylinders of Elliptic Section. 97 



In actually drawing figs. 1-4 the values ofr 2 /a were employed 

 as more convenient for that purpose. 



§ 30. The terms of order z 2 in (14) and (15) indicate that 

 for a given value of x, y the radial displacement and strain are 

 always algebraically greatest when z = 0. Thus the natural 

 influence of rotation in repelling the material from the axis is 

 most felt in the central plane and least in the faces of the disk. 

 The magnitude of the difference is best brought out by 

 reference to the curvature in lines originally parallel to the 

 axis of rotation. Before, however, entering on details it is 

 desirable to consider briefly the reliability of the results, as 

 they depend on terms in a and j3 of order z 2 . 



If the principle of statically equivalent load systems be 

 conceded, it must I think be admitted that the results our 

 solution gives for the curvature are exact, to the present 

 degree of approximation, except at points quite close to the 

 rim. If this be granted it is difficult to see how the curva- 

 ture given by our solution at the rim itself could differ from 

 the true curvature there by a quantity comparable with itself. 

 The difference would have to arise from a difference between 

 the strain of our solution and the true strain extending 

 throughout an area whose thickness is only a few multiples 

 of I. It would also represent not the absolute amount of this 

 difference in the strains of the two cases, but only that part 

 of it which varies with z. Thus an error in our measure of 

 curvature at the rim comparable with that curvature would 

 seem to necessitate the existence either in our solution or in 

 the actual case of large normal stresses varying rapidly 

 with z. Such stresses do not exist in our solution, and they 

 seem incompatible with a free surface at the rim. 



Important evidence on these points is deducible from my 

 previous solution. This gave over the rim stresses 



xx x = C^ 2 , yy x — C 3 e 2 , say, 

 in place of the 



of the present solution. Thus the difference between the 

 strains of the present solution and the previous represents the 

 effects of surface forces 



S 3 =cy 2 /3, ^ 3 =0//3 



at the rim ; while the difference between the strains of the 

 present solution and the true strains represents the effect of the 



surface forces xx 2 , yy 2 given above. The stresses xx 2 , xx s . . . 

 are, it will be noticed, of the same order of magnitude. 

 Phil, Mag. S. 5. Vol. 34. No. 206. July 1892. H 



