66 Lord Kelvin. On the [June 15, 



two balancing couples, as shown in the accompanying diagram. 

 Similarly, we must have four balancing tangential forces S on the 

 four faces parallel to OX, and four tangential forces U on the four 

 faces parallel to OZ. 



15. Considering now an infinitely small change of strain in the 

 cube from (e, /, g. a, &, c) to (e + de, f+df, g + dg, a + da, b + db, 

 c-\-dc) ; the work required to produce it, as we see by considering the 

 definitions of the displacements e,f, g, a, b, c, explained above in 8, 

 is as follows, 



...... (15). 



Hence we have 



P = dw/de ; Q = dwjdf ; R = dw/dg ; 



(16). 



S = dw/da ; T = dw/db ; U = dwjdc ; 



Hence, by (14), and taking L, L to denote linear functions, we find 



(17), 



S = <,, , 



and symmetrical expressions for Q, B, T, U. 



16. Let now our condition of zero strain be one* in which no 

 extraneous force is required to prevent the assemblage from leaving 

 it. We must have P = 0, Q = 0, R = 0, S = 0, T = 0, U = 0, when 

 e = 0, /= 0, g = 0, a = 0, b = 0, c = 0. Hence, by (17), and the 

 other four symmetrical formulae, we see that 



0'W o 0'W , 0'W ., "I 



2^-^ a 8 = 0, .2^-^7/ = 0, 2^-VO, 



> (18)- 



Hence, in the summation for all the points x, y, z, between Avhich 

 and the point O there is force, we see that the first term of the 

 summed coefficients in Q, given by (9) above, vanishes in every case, 

 except those of fg and ea, in each of which there is only a single 

 term ; and thus from (9) and (14) we find 



* The consideration of the equilibrium of the thin surface layer, in these circum- 

 stances, under the influence of merely their proper mutual forces, is exceedingly 

 interesting, both in its relation to Laplace's theory of capillary attraction, and to the 

 physical condition of the faces of a crystal and of surfaces of irregular fracture. 

 But it must be deferred. 



