of a Crystal according to Boscovich. 419 



where Scf)(r) denotes the sum of the values of <j> (r) for the 

 distances between any one point 0, and all the others, and 

 22$ (r) denotes the sum of these sums with the point taken 

 successively at every point of the system. In this double 

 summation cf)(r) is taken twice over, whence the factor J in 

 the formula (10) . 



§ 11. Suppose now the law of force to be such that </>(r) 

 vanishes for every value of r greater than vk, where \ denotes 

 the distance between any one point and its nearest neighbour, 

 and v any small or large numeric exceeding unity, and limited 

 only by the condition that vX is very small in comparison with 

 the linear dimensions of the whole assemblage. This, and 

 the homogeneousness of our assemblage, imply that, except 

 through a very thin surface-layer of thickness v\, exceedingly 

 small in comparison with diameters of the assemblage, every 

 point experiences the same set of balancing forces from 

 neighbours as every other point, whether the system be in 

 what we have called its unstrained condition or in any 

 condition whatever of homogeneous strain. This strain is 

 not of necessity an infinitely small strain, so far as concerns 

 the proposition just stated, although in our mathematical 

 work we limit ourselves to strains which are infinitely small. 



§ 12. Remark also that if the whole system be given as a 

 homogeneous assemblage of any specified description, and if 

 all points in the surface-layer be held by externally applied 

 forces in their positions as constituents of a finite homo- 

 geneous assemblage, the whole assemblage will be in equi- 

 librium under the influence of mutual forces between the 

 points ; because the force exerted on any point by any 

 point P is balanced by the equal and opposite force exerted by 

 the point P' at equal distance on the opposite side of 0. 



§ 13. Neglecting now all points in the thin surface-layer, 

 let N denote the whole number of points in the homogeneous 

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

 homogeneousness of the assemblage, 



22<Kr) = NS^(r) ..... (11), 



and equation (10) becomes 



W = iN2<Mr) (12). 



Hence, by Taylor's theorem, 



BW=^^{(f>'(r)Br + ^"(r)Br 2 } . . . (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 



2F2 



