1893.] Elasticity of a Crystal according to Boscovich. 65 



within it. We have, in 10, by reason of the homogeneousiiess of the 

 assemblage, 



220(r) = N20(r) ................ (11), 



and equation (10) becomes 



W = | N20 (r) .................. (12). 



Hence, by Taylor's theorem, 



" (r) &*} .......... (13) ; 



and using (8) in this, and remarking that if (as in 14 below) we 

 take the volume of our assemblage as unity, so that N is the number 

 of points per unit volume, SW becomes the w of 3 ; we find 



w = 



+ J-0' 00 Q (e, f,g,a, ft, c) + i 



(e* + jfy 2 + flr a + ay* + 6^ 



.... (14). 



14. Let us now suppose, for simplicity, the whole assemblage, in 

 its unstrained condition, to be a cube of unit edge, and let P be the 

 sum of the normal components of the extraneous forces applied to 

 the points of the surface-layer in one of the faces of the cube. The 

 equilibrium of the cube, as a whole, requires an equal and opposite 

 normal component P in the opposite face of the cube. Similarly, let 

 Q and R denote the sums of the normal components of extraneous 

 force on the two other pairs of faces of the cube. Let T be the sum 

 of tangential components, parallel to OZ, of the extraneous forces on 

 either of the YZ faces. The equilibrium of the cube as a whole 

 requires four such forces on the four faces parallel to OY, constituting 



FIG. 1. 









..Y 



JZL- 



\T 



VOL. L1V. 



