of Edinburgh, Session 1872-73. 85 
cannot be introduced into the work referred to. They will appear, 
with extensions, in the second edition (now printing) of the author’s 
Treatise on Quaternions. 
At any point of a strained body, let A, be the vector stress per 
unit of area perpendicular to i, jjl, and v, the same for planes per- 
pendicular to j and k respectively. 
Then, by considering an indefinitely small tetrahedron, we have 
for the stress per unit of area perpendicular to a unit vector to, the 
expression 
ASz’w + fxSjo) + vSko) = — (pit) , 
so that the stress across any plane is represented by a linear and 
vector function of the unit normal to the plane. 
But if we consider the equilibrium, as regards rotation, of an 
infinitely small rectangular parallelepiped whose edges are parallel 
to i, j, k, respectively, we have 
Y(iX +ju + kv) = 0 , 
or 
%Yi(pi = 0 , 
or 
Y.V<pp= 0 . 
This shows that <p is self -conjugate, or, in other words, involves not 
nine distinct constants but only six. 
Consider next the equilibrium, as regards translation, of any 
portion of the solid filling a simply-connected closed space. Let u 
be the potential of the external forces. Then the condition is 
obviously 
fM-Uv)ds+fffdsVu = 0, 
where v is the normal vector of the element of surface ds. 
Here the double integral extends over the whole boundary ot 
the closed space, and the triple integral throughout the whole in- 
terior. 
To reduce this to a form to which the method of my paper on 
Green's and other Allied Theorems {Trans. R.S.E., 1869-70) is 
directly applicable, operate by S.a where a is any constant vector 
whatever, and we have 
JJS.paUvds + fffd&aVu = 0 
