116 Mr C. Chree, The equations of an Isotropic [Oct. 31, 
from some one point. The solution is also applied to a gravi- 
tating spheroid of small excentricity, and to the rotation of a 
sphere or spherical shell. 
Various theories as to rupture are explained, and expressions 
are obtained for the tendency to rupture in accordance with the 
two theories most extensively held. The application of these 
results is fully illustrated in the case of the problems last men- 
tioned. 
In the case of rotation it is found that the tendency to rup- 
ture in a solid sphere is greatest at the centre; while in a thin 
shell it is greatest along the inner surface in the equator, and is 
about four times as great as for a solid sphere of the same radius. 
In the earth, regarded as isotropic, it is found that, on either of 
the theories illustrated, the tendency to rupture due to diurnal 
rotation is greater than the tendency due to forces arismg from 
the earth’s spheroidal shape treated after the method of Professor 
Darwin*. 
A solution is obtained for the vibrations of a sphere or spher- 
ical shell, giving the nature and magnitude of the vibrations due 
to the application of given periodic surface forces. The relation 
between the magnitude of the vibration and that of the constrain- 
ing force, at least when purely normal, is shewn to depend only 
on the degree of the spherical surface harmonic giving the law of 
distribution of the force. By making the surface forces vanish 
equations are obtained giving the frequencies of the various forms 
of free vibration. Perfect agreement exists between the results so 
obtained and those otherwise investigated by Professor Lamb, so 
far as the cases dealt with are the same. 
The equations in cylindrical coordinates are treated in an ana-= 
logous way, and general solutions obtained for the equilibrium or 
vibration of infinite solid or hollow cylinders under the action of 
given bodily or surface forces. It is also shewn how to extend 
these results to the case of coaxial cylindrical layers of different 
materials, and to the treatment of a material whose structure 
varies with the distance from an axis. Expressions for the ten- 
dency to rupture are worked out, and the conditions for their 
application illustrated in the-case of the general solution. 
Two solutions are also obtained applicable to various cases 
where the cylinder is of finite length. Im the one expressions 
are obtained for the strains in ascending powers of z and r, mea= 
sured respectively along and perpendicular to the axis, and this 
is applied to cylinders rotating under various conditions. It is 
found that in general the tendency to rupture in a rotating solid 
cylinder is much less than in a hollow cylinder of the same 
* Phil. Trans. 1879. + Proc. London Mathematical Society, Vol. xv. 
