1892.] Elastic Solids due to given systems of forces. 321 



gravity by an amount equal to the above. When the cylinder is 

 suspended with its axis horizontal, in such a way that bending 

 does not occur, it shortens, while when supported on a smooth 

 horizontal plane in that position it lengthens. The alterations in 

 the mean length in the two cases are given by 



Sl/l = + r/gph/E, 



where rj is Poisson's ratio, while h is the distance of the centre of 

 gravity from the horizontal plane through the points of suspension 

 in the first case and through the points of support in the second. 



When a solid of any shape rotates with uniform angular 

 velocity eo about a principal axis of inertia through its centre of 

 gravity the volume v is increased, the increment being given by 



Bv = co 2 I/3k, 



where i" is the moment of inertia of the body about the axis of 

 rotation. 



W T hen a right cylinder or prism rotates about its axis it 

 shortens, and the mean shortening (— SI) taken over the cross 

 section is given by 



- Bl/l = 7JG) V7^> 



where k is the radius of gyration of the cross section about the 

 axis. 



When a rectangular parallelepiped 2a x 2b x 2c rotates about 

 the axis 2c, the mean increment 28a in the distance between the 

 faces perpendicular to 2a is given by 



8a/a = co 2 p (a 2 - V b 2 )/2E. 



Thus the tendency to increase in length in material lines 

 perpendicular to the axis of rotation becomes reversed when the 

 dimension perpendicular to this and to the axis of rotation is 

 sufficiently great. 



A homogeneous sphere, whether isotropic or aeolotropic, owing 

 to the mutual gravitation of its parts suffers a diminution in 

 volume given by 



— Sv/v = gpRjbk, 



where R is the radius and g "gravity" at the surface. This 

 suffices to prove that the application of the mathematical theory 

 of elasticity to the earth, treated as a homogeneous solid, violates 

 the fundamental condition that the strains must be small, unless 

 the material be assumed to offer a much greater resistance to 

 compression than any known material under normal conditions at 

 the earth's surface. 



The change in volume due to the mutual gravitation in its 

 parts in any very nearly spherical bod} 7 , when isotropic, is shown 



