Page:Scientific Papers of Josiah Willard Gibbs.djvu/468

432 or, writing $$T$$ for the constant $$\frac{aAt}{v},$$ [The terms $$\mp \frac{dV}{dx}k_{0}\gamma_{0}$$ disappear in the algebraic sum since $$\textstyle \sum \displaystyle c_{1}\gamma_{1} = \textstyle \sum \displaystyle c_{2}\gamma_{2}$$. For a similar reason] The first equation makes $$\frac{\textstyle \sum_{0} \displaystyle c_{0}\gamma_{0}}{dx}$$ constant throughout the tube, and since $$\textstyle \sum_{0} \displaystyle c_{0}k_{0}\gamma_{0}'' = \textstyle \sum_{0} \displaystyle c_{0}k_{0}\gamma_{0}',$$ $$\textstyle \sum_{0} \displaystyle c_{0}k_{0}\gamma_{0}$$ must be constant throughout the tube. The second equation then makes $$\frac{dV}{dx}$$ constant throughout the tube. Let $$X = -\frac{dV}{dx}\cdot$$

Our original equation is

Now with $$X$$ constant this is easily integrated. To determine $$H_{0}$$ we have If we put the origin of coordinates in the middle of the tube we have