Solid Cylinders of Elliptic Section. 83 



deduce the simple relation 



R 1 - 1 + B 2 " 1 =2©Vi7(l + i 7 )/E (37) 



The right-hand side of (37) being independent of b or a, we 

 obtain the elegant result 



(R 1 - 1 +E 2 - 1 )/2=l/B, (38) 



where 5 is the radius of curvature at the centre of a face 

 of a circular disk of the same thickness and material as the 

 actual disk rotating with the same angular velocity. It is a 

 well-known theorem in Solid Geometry that the sum of the 

 curvatures at a point on a surface in any two orthogonal normal 

 planes equals the sum of the principal curvatures. We may 

 thus fairly call (Ri _1 +.H 2 ~ 1 )/% the mean curvature, and 

 embody (37) in the following law : The mean curvature pro- 

 duced by rotation at the centre of the faces of an elliptic disk 

 depends solely on the material, thickness, and angular velocity 

 of the disk, being entirely independent of the area or eccen- 

 tricity of the faces. 



§ 16. If a small area at the centre of a face of the disk were 

 polished, the image of a luminous point seen by reflexion 

 from this area would be disturbed by rotation in consequence 

 of the curvature developed. With an elliptic disk the optical 

 effect would be of a somewhat complicated character, and 

 there might be a difficulty in finding any definite quantity to 

 measure, or in interpreting such measurements as could be 

 made. With a circular disk, however, it seems not unreason- 

 able to suppose that measurements might be taken from 

 which the curvature might be deduced. 



In attempting this experiment it would be desirable to 

 employ rapid rotation, so as to produce as large a curvature 

 as possible. At the same time care must be taken that the 

 rotation is kept within the limits prescribed by the linearity 

 of the stress-strain relations, otherwise it would be erroneous 

 to apply the formulae. I thus proceed to obtain formulae 

 connecting the mean curvature at the centre of a face of a 

 disk with the maximum stress-difference S and greatest strain 

 s. Attributing to S and s in these formulae the limiting 

 values consistent with the linearity of the stress-strain rela- 

 tions in the material of the disk, we deduce the largest 

 curvatures it would be possible to produce by rotation suffi- 

 ciently slow for the legitimate application of our formulae. 



For this end let us denote by fi(rj, b/a) and f 2 (y, b/a) the 

 numbers given in Tables I. and II. respectively for the values 



of co^-r- VS//o and &> 2 a-r- V E5//0, answering to given values 



G2 



