86 Proceedings of the Boy at Society 
by taking advantage of the self-conjugateness of <p. This may be 
written 
jQ7tf<S'.Vpa + S.a Yu) = 0 , 
and, as the limits of integration may be any whatever, 
S.V^a + S.aVw = 0 .... (1). 
This is the required equation, the indeterminateness of a rendering 
it equivalent to three scalar conditions 
As a verification, it may be well to show that from this equation 
we can get the condition of equilibrium, as regards rotation , of a 
simply connected portion of the body, which can be written by 
inspection, as 
f/V.ptffyds +ff/Y.pYuds = 0 . 
This is easily done as follows : — (1.) Gives 
S.Vf>cr + S.crVw = 0, 
if, and only if, cr satisfy the condition, 
S.p(V)<r = 0. 
Now this condition is satisfied if 
cr = Yap , 
where a is any constant vector. For 
8 .<p{V)Yap = - S.aV <p(y) p 
= S.aVV^p = 0 . 
Hence 
fffds($.V(pYap + S. apYu) = 0 , 
or 
ff dsS.appUv + fff d&.apYu = 0. 
Multiplying by a, and adding the results obtained by making a in 
succession each of three rectangular vectors, we obtain the required 
equation. 
Suppose cr to be the displacement of a point originally at p, then 
the work done by the stress on any simply connected portion of 
the solid is obviously 
W =JfS.(p(Uv)crds, 
because <p(Uv) is the vector force overcome on the element ds. 
This is easily transformed to 
W = JJfS.Ypcrds . 
