1889.] Mr C. Chree, On Stresses in rotating Spherical Shells. 333 
the torsion modulus is treated also in some detail, and the re- 
sults are on the whole very similar to those given by the uni- 
constant hypothesis. The following remarks are however limited 
to the results deduced from the “ uniconstant” hypothesis. 
Special values—01, 1, -2, 3, °4, °5, ‘6, ‘8, ‘9, and 1—e where e 
is very small—are assigned to the ratio of the radii of the inner 
and outer surfaces. The results obtained for the value ‘01 apply 
to all smaller values of the ratio greater than zero. In each case 
the complete analysis of the state of strain and stress over both 
bounding surfaces is given in tables. The increases in the mean 
radii and the ellipticities of the surfaces are also tabulated. 
The greatest values of the “tendency to rupture” on the “ stress- 
difference” and “ greatest strain” theories are deduced. 
These must be treated as essentially minimum values in con- 
sidering the tendency of the shell to rupture, as the author was 
unable to prove, except in the case of the solid sphere and the 
very thin shell, that greater values might not exist elsewhere. 
The fact however that on either theory the greatest values found 
occur at the centre of the solid sphere, and in the equatorial 
regions of the inner surface of shells of all degrees of thickness, 
renders the existence of greater values on the whole improbable. 
On either theory the existence of a central cavity however 
small approximately doubles the tendency to rupture found in a 
solid sphere. 
To give a clearer idea of the actual magnitude of the pheno- 
mena, the results are also tabulated for the special case of a 
shell possessed of the elasticity of an average piece of iron its 
density being 17/3, its outer radius 4000 miles, and its time of 
rotation 24 hours. The excess of the equatorial over the polar 
diameter of the outer surface increases slowly from 16 miles 
when the sphere is solid to 23 miles when the inner radius is 
1600 miles. The excess then increases much faster as the shell 
grows thinner, reaching 90 miles when the inner radius is 
3600 miles, and 108 miles in the very thin shell. In the solid 
sphere the greatest value of the maximum stress-difference is 
34 tons per square inch. With the existence of a cavity how- 
ever small this rises to 66 tons. It remains nearly constant till 
the radius of the inner surface approaches 1300 miles. It attains 
a maximum of nearly 87 tons per square inch when the radius 
of the inner surface is about 3200 miles, and in the very thin 
shell is about 80 tons. On the greatest strain theory the ten-. 
dency to rupture is greatest in the very thin shell, in which 
case the conclusions of the two theories are identical. Wrought 
iron that can stand a traction of 30 tons per square inch, and 
steel that can stand 60 tons are considered exceptionally strong, 
