A MATHEMATICAL THEORY OF ELASTICITY. 
489 
Cor. 1. If x,y, I, n, ^ denote orthogonal components of a certain strain, and if 
P, Q, R, S, T, U denote components, of the same type respectively, of a stress applied 
to a body while acquiring that strain, the work done upon it per unit of its volume 
Pj7+Q^+R2 + Si+T;}+U^. 
Cor. 2. The condition that two strains or stresses specified by (a;, y, z, ?, n-, 0 and 
{x\y', z', I', ^') in terms of a normal system of types of reference, may be ortho- 
gonal to one another, is 
xx' -\-yy' -f 2 z' + II' + '/iri' + = 0 . 
Cor. 3. The magnitude of the resultant of two, three, four, five, or six mutually 
orthogonal strains or stresses is equal to the square root of the sura of their squares. 
For if P, Q, &c. denote several orthogonal stresses, and F the magnitude of their 
resultant ; and x, y, &c. a set of proportional strains of the same types respectively, 
and r the magnitude of the single equivalent strain, the resultant stress and strain 
will be of one type, and therefore the work done by the resultant stress will be Fr. 
But the amounts done by the several components will be Pj?, Qy, &c., and therefore 
Fr=Pa;+Qj/+&c. 
Now we have, to express the proportionality of the stresses and strains, 
X y r 
Each member must be equal to 
-f &c. 
Vx -f- Qiy -I- &c. ’ 
and also equal to 
Vx -f- Q,y -1- &c. 
x'^ + y'^-^ &c. 
F P2-4-Q,2 I 
Hence - = which gives F^=:P^+Q^+&c., 
F Fr 
and ^~ x^+ ~ y‘^ ' +^c ’ gives r'^=x^ -^y'^ 
Cor. 4. A definite stress of some particular type chosen arbitrarily may be called 
unity ; and then the numerical reckoning of all strains and stresses becomes perfectly 
definite. 
l^ef. A uniform pressure or tension in parallel lines, amounting in intensity to the 
unit of force per unit of area normal to it, will be called a stress of unit magnitude, 
and will be reckoned as positive when it is tension, and negative when pressure. 
Examples. — (1) Hence the magnitude of a simple longitudinal strain, in which lines of the body 
parallel to a certain direction experience elongation to an extent bearing the ratio x to their original 
dimensions, must be called x. 
(2) The magnitude of the single stress equivalent to three simple pressures in directions at right 
angles to one another each unity is — ; a uniform compression in all directions of unity per unit 
of surface, is a negative stress equal to -v/S in absolute value. 
