A MATHEMATICAL THEORY OF ELASTICITY. 
493 
of a unit strain of the first type of the new system ; let (a.^, c^, g^) be the cor- 
responding specification of the second type of the new systen) ; and so on. Then we 
have, for the required formulse of transformation, 
X = -f + a 5^ 5 + «6'^’6 , 
+ ^ 2 ^ 2 + ^ 3 ^ 3 + b^x^+b,x,-\-bQXe, 
^ — glX, -\-g^X^ +^3^3 +g4A’4 ■ 
Example. The transforming equations to pass from a specification {x, y, z, yj, in terms of the 
system of reference used in Examples (6) and (7), Art. X., to a specification cr, rj, ot, co in terms 
of the new system described in Example (3) of Art. IX., and specified in Example (7) of Art. X., 
are as follows : — 
y — + m'co, 
Z — -f w. 
where, as before stated, I, m, n, V, m', n’ are quantities fulfilling the conditions 
P = 1 , 
I -\-m -fra =0, 
Z'^-frai'"^ -fra'^ = l, 
V -f rai' -fra' =0, 
//' -f mm' -f rara' = 0. 
PART II. ON THE MECHANICAL CONDITIONS OF RELATION BETWEEN STRESSES AND 
STRAINS, EXPERIENCED BY AN ELASTIC SOLID. 
Article XIIL — Interpretation of the Differential Equation of Energy. 
In a paper on the Thermo-elastic Properties of Matter, published in the first Number 
of the Quarterly Mathematical Journal (April 1855), it was proved from general 
principles in the theory of the Transformation of Energy, that the amount of work 
{w) required to reduce an elastic solid, kept at a constant temperature, from one 
stated condition of internal strain to another, depends solely on tliese two condi- 
tions, and not at all on the cycle of varied states through which tl)e body may have 
been made to pass in effecting the change, provided always there has been no failure 
in the elasticity under any of the strains it has experienced. Thus for a homogeneous 
solid homogeneously strained, it appears that w is a function of six independent vari- 
ables X, y, z, I, 71, by which the condition of the solid as to strain is specified. 
Hence to strain the body to the infinitely small extent expressed by the variation from 
{x, y, z, I, fi, to (x-}-flfT, y-fdy, z-fdz, the work required to be 
done upon it is 
dw , dw , dw , , dw , dw , , dw 
3 T 2 
