Solid Cylinders of Elliptic Section. 81 



curvature at the rim, will have but a small effect on the 

 limiting safe speed. 



§ 13. Taking into account terms of order I 2 , we find for the 

 corrections 8S and Bs to be added to the values (23) and (24) : 



«a rfpPv(l+v) f « (1-vXl + to,) a A -6 4 } (m 



™- 6(1-7;) I 1 (1 + r;) 2 3a 4 +2a 2 & 2 + 36 4 J'^ ; 



^__ »*pFy(l + v) fi l+3r; a 4 -5 4 ) ^, 



S ~ 6(1 — 17) I 1 + v 3a 4 + 2a 2 & 2 + 3^/•• * ' ^ } 

 According to the theory of statically equivalent load-systems, 

 these corrections should make the values of S and s com- 

 plete so far as terms of order Z 2 , for the measurements of 

 these quantities, being taken at the centre of the disk, are 

 free from the uncertainties which attend the application of 

 our solution to points near the rim. 



The corrections vanish if 77 = 0, and even for ordinary 

 values of 97, such as '25, they are small fractions of co^pl 2 . 

 They are always positive, increasing as the square of the 

 disk's thickness, and are independent of the absolute lengths 

 of the axes of the elliptic section. They are greatest in a 

 circular section. The corrections, if trustworthy, lead to 

 the following law : Unless 77 = 0, the limiting safe speed 

 diminishes as the thickness increases, but this diminution is 

 but trifling while the thickness is a small fraction such as 

 -^q, or even y 1 ^, of the major axis. 



With ordinary values of 97 the corrections would become 

 appreciable if values such as \ or J were assigned to l/a, but 

 the application of our solution to disks as thick as this is not 

 warranted. For thick disks with ordinary values of 77 the 

 only conclusion we can fairly draw is that our solution raises 

 a probability that the limiting speed is less, perhaps con- 

 siderably less, than in a thin disk of the same section and 

 material. 



§ 14. A general idea of the nature of the deformation pro- 

 duced by rotation is most easily derived from a consideration 

 of how the displacement 7 varies throughout the disk. 



According to the first approximation we find from (16) : 



-y = Ms(l-tf 2 /a 1 2 -?/ 2 /6 1 2 ), ....... (29) 



where, for shortness, 



M = 6) 2 / o(a 2 +6 2 )77K/E, (30) 



ai 2 = (a 2 + & 2 ){a 4 + a 2 6 2 (l + 77) +.6 4 }-r {a 4 + a 2 & 2 + 26 4 + ^ 2 (a 2 +36 2 )}, (31) 



V=(a 2 + ^ 2 ){a 4 + a 2 6 2 (l + 77)4-^}^{2a 4 + a^ 2 + 6 4 + 77a 2 (3a 2 + 6 2 )}. (32) 



The value of K is given by (20) . 

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



