494 
PROFESSOR THOMSON’S ELEMENTS OF 
The stress whieh must be applied to its surface to keep the body in equilibrium in 
the state {x, y, z, f], Q must therefore be such that it would do this amount of work 
if the body, under its action, were to acquire the arbitrary strain dx, dy, dz, dl, dn, d^; 
that is, it must be the resultant of six stresses; one orthogonal to the five strains 
dy, dz, d^, dr], d^, and of such a magnitude as to do the work ^dx when the body 
acquires the strain dx ; a second orthogonal to dx, dz, d^, drj, d^, and of such a mag- 
nitude as to do the work '^dy when the body acquires the strain dy ; and so on. If 
a, b, c,f, g, h denote the respective concurrences of these six stresses, with the types 
of reference used in the specification {x, y, z, |, n, Q of the strains, the amounts of the 
six stresses which fulfil those conditions will (Art. XI.) be given by the equations 
1 dw 
a dx’ 
Q 
1 div 
~b dy’ 
„ 1 dw 
-7- ? 
c az 
1 dw 
fW 
T: 
1 dw 
~9 df] ’ 
\ dw ^ 
^—hTxf 
and the types of these component stresses are determined by being orthogonal to the 
fives, of the six strain-types wanting the first, the second, &c. respectively. 
Cor. If the types of reference used in expressing the strain of the body constitute 
an orthogonal system, the types of the component stresses will coincide with them, 
and each of the concurrences will be unity. Hence the equations of equilibrium of 
an elastic solid referred to six orthogonal types are simply 
11 
,, dw 
II 
^1 S' 
1 
_ dw 
11 
II 
Article XIV. — Reduction of the Potential Function, and of the Equations of 
Equilibrium, of an Elastic Solid to their simplest Forms. 
If the condition of the body from which the work denoted by w is reckoned be 
that of equilibrium under no stress from without, and x,y, z, |, rj, ^ be chosen each 
zero for this condition, we shall have, by Maclaurin’s theorem, 
w—Wfx, y, z, I, n, (:^) + llfx,y, z, rj, ^)-l-&c., 
where Hg, H 3 , &c. denote homogeneous functions of the second order, third order, 
&c. respectivelv. Hence &c. will each be a linear function of the strain- 
dx dy 
coor<linates, together with functions of higher orders derived from H 3 , &c. But ex- 
perience shows that within the elastic limits, the stresses are very nearly if not quite 
|)roportional to the strains they are capable of producing; and therefore H 3 , &c. 
may be neglected, and we have simply 
w=U.,{x, y, z, I, ri, Q. 
