116 
Proceedings of Roigal Society of Edinburgh. [sess. 
We may from this form of w at once go back to that in terms of 
i j/ by means of equation (8) above. Thus, 
W — ( 53 ), 
or, again, by equation (7), 
w = ^2>x4) (54). 
Hence by substituting - £S for ij/£ v we get 
-is - s MitAtvUv'I'Qh 
and .*. by p. 104 
77*o = 2£w(£, * 0 ,^, 1 /^). ..... (55), 
or, again, by equation (8), 
27*0 = 2£w>(£,*o, V (56). 
The equation (23) of equilibrium thus becomes in this case 
3 + 2£w(£,A, V = 0 (57) 
Thus in the case of isotropic bodies in which c is the compressi- 
bility and n the rigidity, we have {Mess, of Math., vol. xiv. pp. 30, 
31) 
= 2nij/o) ~(f ~ (58), 
.... (59). 
27co 
(60). 
6n 9c 
Thus from these equations and equation (49) 
W= S’W 
Again, from the first of these equations and equation (8) we have 
( C 71 \ 
V)= -nSV 1 >pT ll + {^ -j)S 2 Vr], 
or 
2 w = rcS V 1% S V iVl + reS V 1 V 2 S W2 + (c - -|»)s 2 V v . (61 ), 
and from equation (56) 
27*o = — wSoo V . rj - n V l S*o77 1 — {m — ?^)*oS V rj . . (62), 
