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To determine the relations that must exist between forces that act on a homogeneous elastic solid body and the deformations of its elements so that equilibrium is maintained, we will apply the following principle of Lagrange: In order that a system, whose virtual motions are interchangeable, is in equilibrium, it is necessary and sufficient that the mechanical work done from forces in any virtual motion is equal to zero.

Let X, Y, Z be the components of accelerating forces that act on every point of the body; L, M, N the components of the force acting on every point of its surface; and r constant density. We associate a virtual motion u, v, w with every point of the deformed body, and we denote by δu,δv, δw the variations this motion will take. Clearly, the work done by this motion by the given forces will be

where S is the volume occupied by the body and σ is its surface. The work done by elastic forces will be equal to the increase in potential of the whole body, that is,

Whence from the principle of Lagrange

Now, denoting by α, β, γ the cosines of the angles that the normal directed toward the interior of the body forms with the positive axes,

Substituting in it equation 11 and setting the coefficients δu, δv, δw of the triple and double integrals equal to zero separately, the equations of equilibrium are obtained:

To these equations, we will add the others which express

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