A MATHEMATICAL THEORY OF ELASTICITY. 
491 
where I, m, n, V, m', n' denote quantities fulfilling the following conditions 
P +7rP +jp 
I -\-m +n =0, 
W +mm' -{-nn' = 0, 
nr 
-f- n'^ = 1 
(8) If 
I' +m' +n' =0. 
(1 - 2eP)X2 + (1 - 2eQ) Y2 + ( 1 ~ 2eR)Z2 - 2e ^/2{SYZ + TZX + UXY) = 1 
and 
be the equation of the ellipsoid representing a certain stress, the amount of work done by this stress, 
if applied to a body while acquiring the strain represented by the equation in the preceding example, 
P^ + Q2/ + R^ + S? + T^ + U?. 
Cor. Hence, if the variables X, Y, Z be transformed to any other set (X', Y', Z') fulfilling the con- 
dition of being the coordinates of the same point, referred to another system of rectangular axes, 
the coefficients x, y, z. See., y^, z^, &c. in two homogeneous quadratic functions of three variables, 
(1 - 2a?) X2 -j- (1 - 2y) Y2 -h ( 1 - 2^) Z2 - 2 v/2 YZ + >iZX + ?X Y) 
{l-2x,)X^ + {l- 2y,)Y^ + ( 1 - 2z,)Z^ - 2V2 (?,YZ + >,^ZX -f ^^X Y), 
and the corresponding coefficients x', y', z', Sec., x^, y], z\. Sec. will be so related that 
ay'a?' -j- y'y! + + Wt + + »)»?/ + ??/ i 
or the function + + + + + of the coefficients is an “invariant” for linear transforma- 
tions fulfilling the conditions of transformation from one to another set of rectangular axes. Since 
x + y-\-z and a?, -{- -f are clearly invariants also, it follows that AA^ -f -|- CC, -f 2DD; -|- 2EE, -H 2FF, 
is an invariant function of the coefficients of the twm quadratics 
AX^ + BY2 -p CZ^ + 2DYZ q- 2EZX -f 2FXY 
and A;X2 -f B^Y2 -f- C^Z^ + 2D^YZ -|- 2E^ZX -p 2F,X Y, 
which it is easily proved to be by direct transformation. 
Article XI. — On Imperfect Concurrences of two Stress or Strain-types. 
Def. The concurrence of any stresses or strains of two stated types, is the propor- 
tion which the work done when a body of unit volume experiences a stress of either 
type while acquiring a strain of the other, bears to the product of the numbers 
measuring the stress and strain respectively. 
Cor. 1. In orthogonal resolution of a stress or strain, its component of any stated 
type is equal to its own amount multiplied by its concurrence with that type; or the 
stress or strain of a stated type which, along with another or others orthogonal to it 
have a given stress or strain for their resultant, is equal to the amount of the given 
stress or strain reduced in the ratio of its concurrence with that stated type. 
Cor. 2. The concurrence of two coincident stresses or strains is unity; or a per- 
fect concurrence is numerically equal to unity. 
Cor. 3. The concurrence of two orthogonal stresses and strains is zero. 
Cor. 4. The concurrence of two directly opposed stresses or strains is —1. 
3 T 
MDCCCLVI. 
