Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/99

Rh 186. If we have where $$\Gamma$$ is complete, the integral equation will be of the form  For this gives   and $$\alpha$$ and $$\beta$$ may be determined so as to satisfy the equations  187. The differential equation will be satisfied by  whence   If $$\Gamma$$ is complete, the constants $$\alpha$$ and $$\beta$$ may be determined to satisfy the equations  188. If where $$\Gamma^2 - \Lambda^2$$ is a complete dyadic, and we may set {{MathForm2||$$\rho = \{ \tfrac{1}{2} e^{t \Gamma} + \tfrac{1}{2} e^{-t \Gamma} + \cos \{ t \Lambda \} - \text{I}. \alpha + \{ \tfrac{1}{2} e^{t \Gamma} - \tfrac{1}{2} e^{-t \Gamma} + \sin \{ t \Lambda \} \}. \beta$$}} which gives