A MATHEMATICAL THEORY OF ELASTICITY. 
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sented by the ellipsoid 
(1 + A + A')X2 + (1 + B + B') Y2 + (1 + C + C')Z2 + (D + D') YZ + (E + E')ZX + (F + F')X Y = 1 . 
Article VII 1 . — Specification of Sir aim and Stresses. 
Prop. Six stresses or six strains of six distinct arbitrarily chosen types niay be 
determined to fulfil the condition of having a given stress or a given sti’ain for their 
resultant, provided those six types are so chosen that a strain belonging to any one 
of them cannot be the resultant of any strains whatever belonging to the others. 
For, just six independent parameters being required to expi-ess any stress or strain 
whatever, the resultant of any set of stresses or strains may be made identical with 
a given stress or strain by fulfilling six equations among the parameters which they 
involve; and therefore the magnitudes of six stresses or strains !)elonging to the six 
arbitrarily chosen types may be determined, if their resultant be assumed to be iden- 
tical with the given stress oi- strain. 
Cor. Any stress or strain may be numerically specified in terms of numbers 
expressing the amounts of six stresses or strains of six arbitrarily chosen types which 
have it for their resultant. 
Types arbitrarily chosen for this purpose will be called types of reference. The 
specifying elements of a stress or strain will be called its components according to 
the types of reference. The specifying elements of a strain may also be called its 
coordinates, with reference to the chosen types. 
Examples. — (1) Six strains in each of which one of the six edges of a tetrahedron of the solid is 
elongated while the others remain unchanged, may be used as types of reference for the specification 
of any kind of strain or stress. The ellipsoid representing any one of those six types will have its 
two circular sections parallel to the faces of the tetrahedron which do not contain the stretched side. 
(2) Six strains consisting, any one of them, of an infinitely small alteration either of one of the 
three edges, or of one of the three angles between the faces, of a parallelepiped of the solid, while 
the other five angles and edges remain unchanged, may be taken as types of reference, for the 
specification of either stresses or strains. In some cases, as for instance in expressing the probable 
elastic properties of a crystal of Iceland spar, it may be convenient to use an oblique parallelepiped 
for such a system of types of reference; but more frequently it will be convenient to adopt a system 
of types related to the deformations of a cube of the solid, in the manner described. 
(3) If AX2 + BY2 + CZ2 + DYZ-fEZX + FXY=1 
be the equation of the surface of a portion of the solid referred to oblique or rectangular coordi- 
nates, we may take the six strains, in any one of which the same portion of the solid becomes 
altered in shape to a surface whose equation differs from the preceding only in having one of the six 
coefficients altered by an infinitely small quantity, as six types of reference for specifying stresses 
and strains in general. 
Article IX. — Orthogonal Types of Reference. 
Def. A normal system of types of reference is one in which the strains or stresses 
of the different types are all six mutually orthogonal (fifteen conditions). A normal 
