26 
Proceedings of Royal Society of Rdinhurgli, [dec. 5, 
Hence if E denotes its magnitude per unit of volume ; or the 
potential energy of unit volume in the condition (a, y) reckoned 
from zero in the condition (1, 1, 1) ; we have 
E = i/^(a, fty) ...... (13), 
where xp denotes a function of which the magnitude is unaltered 
when the values of a, /?, y are interchanged. Consider a portion 
of the solid, which, in the unstrained condition, is a cube of unit 
side, and which in the strained condition (a, y), is a rectangular 
parallelopiped Ja. Jy. In virtue of isotropy and symmetry, 
we see that the pull or pressure on each of the six faces of this 
figure, required to keep the substance in the condition (a, /3, y), is 
normal to the face. Let the amounts of these forces per unit area, 
on the three pairs of faces respectively, he A, B, C, each reckoned 
as positive or negative according as the force is positive pull, or 
positive pressure. We shall take 
A + B + C = 0 (14); 
because normal pull or pressure uniform in ail directions produces 
no effect, the solid being incompressible. The work done on any 
infinitesimal change from the configuration (a, /?, y), is 
A J(fiy)d( Ja) + BV(ya)^(V/3) + C J(ap)d{Jy) , ] 
or (because a/?y=l) 
1-. . (15). 
=7 J 
10. Let 8a, 8^, 8y be any variations of a, J3, y consistent with (2): 
so that we have 
(a + 8a) {/3 + 8/3) (y + 8y) = 1 
and a/3y = 1 
} . . . . (16). 
How suppose 8a, 8/?, 8y to be so small that we may neglect their 
cubes and corresponding products, and all higher products. We 
have 
~ ^ + a8j38y + /?8y8a + y8a8/5 = 0 . . (17), 
a Id y 
whence 
whence, and by the symmetrical expressions 
