Solid Cylinders of Elliptic Section. 



85 



the disk and all sections parallel to them remain plane, and 

 every point retains unaltered its distance from the central 

 plane. All lines originally parallel to the axis of rotation 

 remain parallel to that axis. 



§19. The value of 7 at the centre of the face z = l will here 

 be termed 81. The quantity ( — 281) measures the reduction 

 in the axial thickness, — i. e. the thickness of the disk measured 

 along the axis of rotation. Since (29) agrees with (16) at 

 the faces of the disk, we have 



(-8l/l) = M (42) 



The following table shows how this quantity depends on 

 the shape of the disk and on the value of 77. 



Table III. 

 Value of {-$l/l) + (a>*pa*/E). 



b/a. 



t]= 0. 



•2. 



•25. 



•3. 



•5. 











•06 



•083 



•1 



•16 



•2 







•0708 



•0886 



•1066 



•1790 



•4 







•0832 



•1046 



•1264 



•2161 



•6 







•1034 



•1307 



•1586 



•2763 



•8 







•1297 



•1645 



•2002 



•3527 



1-0 







•16 



•2031 



•2475 



•4375 



When less than four figures are given the result is exact. 

 For disks of given material, possessed of a given major axis 

 and angular velocity, the reduction in axial thickness increases 

 as the minor axis increases. For different materials, if we 

 were to regard o> 2 pa 2 /E as constant, we should conclude that 

 the reduction in axial thickness in any form of elliptic disk 

 increased with Poisson's ratio. 



When a disk is rotating at its limiting speed, its reduction 

 in axial thickness may be derived by combining Table III. 

 with Table I. or Table II., allowing to S or s its limiting value 

 for the material. Thus, let/ x and/ 2 be the same functions as 

 before, and let fs(rj, b/a) represent the numerical values in 

 Table III.; also denote by ( — 81JI) and ( — 8l 2 /l) the reduc- 

 tions per unit of axial thickness answering to the maximum 

 stress-difference S and the greatest strain s respectively. 

 Then we have 



{-SIS =(§/E)x \M Vl b/a) }* x/ 3 ( % 5/a), . . . (43) 

 (Sk/l)= sx. \Mn,tya)\ 2 xMv,l>/a). . . . (44) 



