Elastic Solids to Metrology. 557 



force systems, by suffixes, (2) becomes in the absence of bodily 

 forces 



3^t?=jJ(F^ + G# + Hi*)**! + jj (F 2 * + G 2 y + H 2 z)dS 2 + . . (60) 



As each of the surface integrals is self-contained, we see 

 that if a given system of forces act over a given surface, 

 the total change produced in the volume of the material is 

 independent of the existence of cavities. For instance, if 

 given forces act over an outer spherical surface, the change 

 in the total volume of the material within it is the same 

 whether the sphere be absolutely solid, or whether it contain 

 a cavity or cavities over whose surfaces no forces act. If 

 there is a large cavity, the mean change of volume per unit 

 volume is correspondingly greater. 



If a uniform pressure p x act over a surface S l5 the 

 corresponding change of volume is given by 



3&Si'= —p^X^ + fiiy + Viz)dS } — — Zp x v u 



hv=^ Pl v l jk, (61) 



where v 1 is the volume of the space enclosed by Si. 



If, then, uniform pressure of given intensity be applied 

 over the surface of a cavity, the total change of volume in 

 the material depends only on the volume of the cavity, and is 

 independent of its form or position, or of the extent of the 

 solid, supposing it bounded by free surfaces. 



§ 17. There are also some simple general properties de- 

 ducible from (2) in the case of isotropic material acted on 

 only by bodily forces following the gravitational law. 



Thus, suppose a centre of gravitational force at the origin 

 of coordinates, and so 



X/^=Y/y=Z/ < 8=-m/r 3 , . . . («2) 



where m is a constant. The corresponding elastic increment 

 in the volume of the solid is by (2) 



Sc = - (m/U) RSfi^-Vfa dy dzl, . . (63) 



the integral extending throughout the whole solid. 



But the integral, within the square brackets, simply repre- 

 sents the potential at the origin answering to a distribution 

 of unit density throughout r. Thus the change in volume of 

 the solid varies as the value of its potential at the centre 

 of force, and can be calculated when the potential can. 

 If, for instance, the solid were a sphere whose centre was at 

 a distance R from the centre of force, we should have 



8v/v=-7n/(MR) (64) 



