<54 Lord Kelvin. On the [June 15, 



and corresponding symmetrical expressions for the other fifteen 

 coefficients. 



10. Going back now to 3, let us find 10, the work per unit 

 volume, required to alter our homogeneous assemblage from its un- 

 strained condition to the infinitesimally strained condition specified 

 by e,/, #, a, 6, c. Let 0(r) be the work required to bring two points of 

 the system from an infinitely great distance asunder to distance r. 

 This is what I shall call the mutual potential energy of two points at 

 distance r. What I shall now call the potential energy of the whole 

 system, and denote by W, is the total work which must be done to 

 bring all the points of it from infinite mutual distances to their actual 

 positions in the system ; so that we have 



(10), 



where 20 (r) denotes the sum of the values of 0(r) for the distances 

 between any one point O, and all the others; and 220 (r) denotes 

 the sum of these sums with the point O taken successively at every 

 point of the system. In this double summation (r) is taken twice 

 over, whence the factor f in the formula (10). 



11. Suppose now the law of force to be such that 0(r) vanishes 

 for every value of r greater than ^X, where X 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 v\ is very small in comparison with the linear dimensions of 

 the whole assemblage. This, and the homogeneousness of our assem- 

 blage, imply that, except through a very thin surface layer of thick- 

 ness v\ 9 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 homo- 

 geneous assemblage of any specified description, and if all points in 

 the surface-layer be held by externally applied forces in their posi- 

 tions as constituents of a finite homogeneous assemblage, the whole 

 assemblage will be in equilibi'ium 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 



