320 Mr Chree, Changes in the dimensions of [Mar. 7, 



Professor Betti, but he does not seem to have considered the 

 general case, nor to have made applications such as those treated 

 here. From the formulae for the mean strains the change can be 

 found in the mean length, taken over the cross section, of any 

 right cylinder or prism subjected to any given system of forces. 

 Similarly the change in the whole volume of any elastic solid of 

 any shape can always be expressed as the sum of a volume and a 

 surface integral involving only the applied forces and the elastic 

 constants of the material. 



Thus in an isotropic solid acted on by bodily forces whose 

 components are X, Y, Z per unit volume, and by surface forces 

 whose components are F, G, H per unit surface, the change Bv in 

 the volume is given by 



SkBv = fJJ(Xcc + Yy + Zz) dxdydz + JJ(Fx + Gy + Hz) dS, 



where k denotes the bulk modulus, or m—^n in Thomson and 

 Tait's notation. It is obvious from the equations of statical equi- 

 librium that the position of the origin in the above expressions is 

 immaterial. In any homogeneous aeolotropic solid the change in 

 volume may be similarly determined, but the expressions under 

 the integral signs are a little longer. 



The several formulae both for isotropic and aeolotropic solids 

 are applied to a variety of special cases, a few of which will serve 

 for illustration. The material to which the following results apply 

 is, unless otherwise stated, assumed isotropic. 



When a solid of any shape is suspended from a point, or a 

 series of points in one horizontal plane, its volume v is greater 

 than if "gravity" did not act, and the increment Bv due to 

 '•'gravity", represented by g, is given by 



Bv/v = gphjSJc, 

 where p is the density and h the distance of the centre of gravity 

 below the point, or points, of suspension. On the other hand, if a 

 body be supported on a smooth plane, or at a series of points in a 

 horizontal plane, its volume is diminished owing to the action of 

 gravity, the diminution (— Bv) being given by 



-Bvjv=gph'/3Ic, 

 where h' is the height of the centre of gravity above the plane of 

 support. 



When a right cylinder or prism is suspended with its axis 

 vertical its length I is increased, and the mean increment Bl taken 

 over its cross section is given by 



8l/l=gpl/2E, 

 where E is Young's modulus. When the cylinder rests on a 

 smooth horizontal plane with its axis vertical, it shortens under 



