418 Lord Kelvin on the Elasticity 



Now by (4) we have 



x8x + ySy + zSz = ex 2 +fy 2 + gz 2 + ayz -f bzx + cxy . . (6) , 

 and 

 Bx 2 + fy 2 + Sz 2 = e 2 x 2 +f 2 y 2 + 9 2 z 2 



+ i[a 2 (y 2 + z 2 ) + b 2 {z 2 + x 2 )+c 2 {x 2 +y 2 )] 

 + libc + (f+g)a-]yz+[ica+ (g + e)b]zx 

 + [±ab+(e+f)c]xy (7). 



Using (6) and (7) in (5), we find 



ov 



— = r~ 2 (ex 2 +fy 2 +gz 2 + ayz + bzx + cxy) + Q(<?,/, g, a, &, c) (8) , 



where Q denotes a quadratic function of e, f, &c, with 

 coefficients as follows : — 



Coefficient of \e 2 is 



X" X* 



r 2 7 



» 



?J 



2" 55 





?j 



?> 



/# 55 



fz 2 



55 



55 



be „ 



4 ^ r4 



» 



?5 



ea „ 





?J 



55 



eh „ 





(9), 



and corresponding symmetrical expressions for the other 

 fifteen coefficients. 



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

 unit volume, required to alter our homogeneous assemblage 

 from its unstrained condition to the infinite si m ally strained 

 condition specified by e, f, g, a, b, c. Let cj> (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 



W = *SUM (10), 



