Solid Cylinders of Elliptic Section. 7 1 



through the centre of gravity. Amongst the cases treated 

 was that of a thin elliptical disk. The solution obtained for 

 this case * was, as explicitly stated in my paper, only approxi- 

 mate, certain surface conditions not being exactly satisfied. 

 In a criticism of solutions hitherto proposed for thin rotating 

 circular disks, Professor Pearson t referred to mine in language 

 which, though comparatively flattering, implied doubts as to 

 its trustworthiness, on the ground that it did not satisfy certain 

 surface conditions he regarded as essential. I have since 

 shown that my method can supply a solution J for a thin 

 circular disk which satisfies all the conditions held essential 

 by Professor Pearson, and that this only adds to my original 

 expressions for the displacements certain terms of the third 

 'power of the thickness. The first object of the present paper 

 is to show that a similar unimportant addition meets Professor 

 Pearson's objection in the more general case of an elliptic 

 disk. 



My principal reason, however, for returning to the subject 

 is that in my previous paper no attempt was made to evolve 

 the physical conclusions latent in the somewhat complicated 

 mathematical formulae. This deficiency will, it is hoped, be 

 met by the present paper, more especially by the tables of 

 numerical results. Very likely the mathematical problem 

 never has its conditions exactly realized in practice, but its 

 solution may nevertheless prove a useful guide and auxiliary 

 to experiment. 



In treating elastic solids whose surfaces consist of different 

 portions intersecting at finite angles, it has in general been 

 found impossible to satisfy all the conditions which the 

 ordinary theory regards as holding at every point of a surface 

 between the stresses in the material and the applied forces. 

 Saint- Venant § , however, and other eminent authorities, have 

 held that when a dimension of a body is very small, as the 

 thickness in a thin disk, all that is required, at least for 

 practical purposes, is an equality between the statical resultant 

 of the stresses and that of the surface forces applied over the 

 small dimension. The applied forces may in fact be replaced 

 by any statically equivalent system, and, according to the 

 authorities above referred to, the displacements given by the 

 mathematical theory which so replaces the actually existing 



* Quarterly Journal, I. c. pp. 27-28, equations (125)-(128). 



t ' Nature,' vol. xliii. (1891), p. 488. 



| Cambridge Philosophical Society's i Proceedings/ vol. vii. pp. 201- 

 215, 1891. 



§ See Pearson's ' Elastical Researches of Barre" de Saint- Venant,' arts. 

 8 and 9. 



