558 Applications of Elastic Solids to Metrology. 



It is clear from (63) that the change of volume is unaltered 

 by moving the centre o£ force from any one point to any 

 other point which lies on the same equipotential surface of 

 the solid. 



In applying (63) it must be borne in mind that in general 

 a body cannot be in equilibrium under the action of a single 

 centre of force, and that when equilibrium exists under the 

 joint action of a number of centres of force it is to a certain 

 extent arbitrary to regard the effect of one as independent of 

 the others. In the case of a planet, for instance, the " cen- 

 trifugal force " in the orbital path balances the sun's 

 attraction. Jf we regarded the planet as an elastic solid 

 describing a circular path, and neglected for the time being 

 the effects of self-gravitation and rotation about an axis, we 

 should find its elastic change of volume given by two terms. 

 One of these terms would be of the type (64), depending on 

 the mass of the sun ; the other would depend on the angular 

 velocity in the orbit. The persistence, however, of the 

 planet in its orbit implies a necessary relation between the 

 gravitational and " centrifugal " forces, so that the two terms 

 in the expression for Sv could be combined into one, con- 

 taining as we choose the angular velocity or the gravitational 

 constant. 



§ 18. In obtaining (63) we treated the source of the 

 gravitational forces as wholly external to the solid. In a 

 self-gravitating homogeneous spherical " earth " of radius a, 

 we should have 



X/*=Y/</ = Z/-= -</?/«, 



where g' is " gravity " at the surface. Substituting in (2) 

 we easily deduce 



Sv/v=—g'pal5k (65) 



As I have repeatedly remarked elsewhere, the application 

 of such a formula to the actual earth — or other principal 

 planet — leads to numerical results which cannot be reconciled 

 with the fundamental hypotheses of mathematical elasticity 

 unless k be much larger than in known materials under the 

 conditions existing near the earth's surface. 



In small bodies, on the other hand, the effects of self- 

 gravitation are very small. For instance, for a sphere of 

 iron (p = 7'5, £=15xl0 8 grammes wt. per sq. cm.) of one 

 metre radius we should get from (65) 



fo/t>=— 2*10- u , 



or 8a= — 7 x 10~ 12 mm. 



r To be continued.] 



