Applications of Elastic Solids to Metrology. 533 



When the material is aeolotropic but symmetrical with 

 respect to 3 planes — supposed parallel to the coordinate 

 planes — the suffixes 1, 2, 3 distinguish the directions parallel 

 respectively to the axes of x, y, and z. Thus Ej denotes 

 Young's modulus for tension parallel to the #-axis, while 

 ^12 i = Vn) is Poisson's ratio when the tension and the 

 corresponding strain are parallel, the one to x the other to y. 

 In all cases r represents the total volume of the solid, p its 

 density, X, Y, Z the components of the bodily and F, Gr, H 

 of the surface forces ; A is the dilatation, a, ft, y, the elastic 

 displacements, and e,f, g (=da/dx, &c.) the expansion strains. 

 Mean values are distinguished by a horizontal line, e. g. e, 

 while 8 denotes (elastic) change in a dimension or in volume. 



For an isotropic material the general formulse obtained in 

 the paper referred to above were of the types : 



Evg=E\\\^dxdydz 



= $\Zz- V (Xx + Yy) }dxdydz+§{Kz- v (Fx+Gy)\dS, (1) 



U8v =j]j( Kx + Yy + Z*) dxdydz + jJ(F* + Qy + Uz)dS ; . . (2) 



where the triple integrals are taken throughout the entire 

 volume of the solid, the double integrals over its whole 

 surface or surfaces. The coordinate axes are rectangular, 

 the origin being at any convenient point in the solid. For 

 aeolotropic solids there are analogous equations, for which 

 reference may be made to the original paper. 



Suspended and Supported Solids. 



§ 2. Equation (2) gives the increment 8v in the total 

 volume, and thence the change in the mean density ; while 

 (1) gives the change in the mean length of a right cylinder. 

 Thus, suppose a cylindrical bar of any form of section a hung 

 up by an end so that its axis — taken as axis of z — is vertical. 

 We then have, measuring z downwards, 



X=Y=0, Z=gp, 



where g is " gravity." If we neglect the pressure of the 

 surrounding medium, F, G, H vanish everywhere except at 

 the top of the cylinder, where H answers to the tension of 

 the string or support. We may avoid the necessity of con- 

 sidering this tension by simply supposing the origin of 

 coordinates in the upper end of the bar ; for then z, and so 

 H^, vanishes everywhere where H itself does not vanish. If 

 then SI represent the mean change of the length I (i. e. the 

 Phil. Mag. 8. 6. Vol. 2. No. 11. Nov. 1901. 2 N 



