as in the ordinary calculus, but we must not apply these equations to cases in which the values of are not homologous.
183. If, however, is any constant dyadic, the variations of will necessarily be homologous with and we may write without other limitation than that is constant,
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(1)
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(2)
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(3)
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(4)
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A second differentiation gives
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(5)
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(6)
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(7)
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184. It follows that if we have a differential equation of the form
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the integral equation will be of the form
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representing the value of for For this gives
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and the proper value of for
185. Def.—A flux which is a linear function of the position-vector is called a homogeneous-strain-flux from the nature of the strain which it produces. Such a flux may evidently be represented by a dyadic.
In the equations of the last paragraph, we may suppose to represent a position-vector, the time, and a homogeneous-strain-flux. Then will represent the strain produced by the flux in the time
In like manner, if represents a homogeneous strain, will represent a homogeneous-strain-flux which would produce the strain in the time