A MATHEMATICAL THEORY OF ELASTICITY. 
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amount per unit of area) experienced by the enclosed sphere of the solid, at the 
different parts of its surface, when subjected to the stress. 
Def. A stress and an infinitely small strain so related are said to be of the same 
type ; and the ellipsoid, by means of which the distribution of force over the surface 
of a sphere of unit radius is represented in one case and the displacements of particles 
from the spherical surface are shown in the other, may be called the geometrical 
type of either. 
Article V. — Conditions of Perfect Concurrence between Stresses and Sti'ains. 
Def Two stresses are said to be coincident in direction, or to be perfectly con- 
current, when they only differ in absolute magnitude. The same relative designations 
are applied to two strains differing from one another only in absolute magnitude. 
Cor. If two stresses or two strains differ by one being reverse to the other, they 
!nay be said to be negatively coincident in direction ; or to be directly opposed or 
directly contrary to one another. 
Def. When a homogeneous stress is such that the normal component of the mutual 
force between the parts of the body on the two sides of any plane whatever through 
it is proportional to the augmentation of distance between the same plane and 
another parallel to it and initially at unity of distance, due to a certain strain expe- 
rienced by the same body, the stress and the strain are said to be perfectly concur- 
rent ; also to be coincident in direction. The body is said to be yielding directly to a 
stress applied to it, when it is acquiring a strain thus related to the stress ; and in 
the same circumstances, the stress is said to be working directly on the body, or to 
be acting in the same direction as the strain. 
Cor. 1. Perfectly concurrent stresses and strains are of the same type. 
Cor. 2. If a strain is of the same type as a stress, its reverse will be said to be 
negatively of the same type, or to be directly opposed to the strain. A body is said 
to be working directly against a stress applied to it when it is acquiring a strain 
directly opposed to the stress ; and in the same circumstances, the matter round the 
body is said to be yielding directly to the reactive stress of the body upon it. 
Article VI. — Orthogonal Stresses and Strains. 
Def. 1. A stress is said to act right across a strain, or to act orthogonally to a 
strain, or to be orthogonal to a strain, if work is neither done upon nor by the body 
in virtue of the action of the stress upon it while it is acquiring the strain. 
fle/. 2. Two stresses are said to be orthogonal when either coincides in direction 
with a strain orthogonal to the other. 
Def 3. T 'wo strains are said to be orthogonal when either coincides in direction 
with a stress orthogonal to the other. 
Examples. — (1) A uniform cubical compression, and any strain involving no alteration of volume, 
are orthogonal to one another. 
3 s 2 
