Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/98

82 as in the ordinary calculus, but we must not apply these equations to cases in which the values of $$\Phi$$ are not homologous.

183. If, however, $$\Gamma$$ is any constant dyadic, the variations of $$t \Gamma$$ will necessarily be homologous with $$t \Gamma$$ and we may write without other limitation than that $$\Gamma$$ is constant,    A second differentiation gives    184. It follows that if we have a differential equation of the form the integral equation will be of the form  $$\rho '$$ representing the value of $$\rho$$ for $$t = 0.$$ For this gives  and the proper value of $$\rho$$ for $$t = 0.$$

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 $$\rho$$ to represent a position-vector, $$t$$ the time, and $$\Gamma$$ a homogeneous-strain-flux. Then $$e^{t \Gamma}$$ will represent the strain produced by the flux $$\Gamma$$ in the time $$t.$$

In like manner, if $$\Lambda$$ represents a homogeneous strain, $$\{ \log \Lambda \} / t$$ will represent a homogeneous-strain-flux which would produce the strain $$\Lambda$$ in the time $$t.$$