# Zur Theorie des Krümmungsmasses

• Königlichen Gesellschaft der Wissenschaften

## Zusammenfassung

Die Bedingungsgleichung, dass
$$pd{x^2} + 2qdxdy + rd{y^2}$$
das reine Product aus zwei vollständigen Differentialen sei, ist folgende:
$$\begin{gathered} {\text{ }}0 = 2(qq - pr)\left( {\frac{{\partial \partial p}}{{\partial {y^2}}} - 2\frac{{\partial \partial q}}{{\partial x\partial y}} + \frac{{\partial \partial r}}{{\partial {x^2}}}} \right) + p\left( {\frac{{\partial p}}{{\partial y}} \cdot \frac{{\partial r}}{{\partial y}} - 2\frac{{\partial q}}{{\partial x}} \cdot \frac{{\partial r}}{{\partial y}} + \frac{{\partial r}}{{\partial x}} \cdot \frac{{\partial r}}{{\partial x}}} \right) \hfill \\ + q\left( {\frac{{\partial p}}{{\partial x}} \cdot \frac{{\partial r}}{{\partial y}} - \frac{{\partial p}}{{\partial y}} \cdot \frac{{\partial r}}{{\partial x}} + 4\frac{{\partial q}}{{\partial x}} \cdot \frac{{\partial q}}{{\partial y}} - 2\frac{{\partial q}}{{\partial x}} \cdot \frac{{\partial r}}{{\partial x}} - 2\frac{{\partial p}}{{\partial y}} \cdot \frac{{\partial q}}{{\partial y}}} \right) + r\left( {\frac{{\partial p}}{{\partial x}} \cdot \frac{{\partial r}}{{\partial x}} - 2\frac{{\partial p}}{{\partial x}} \cdot \frac{{\partial q}}{{\partial y}} + \frac{{\partial p}}{{\partial y}} \cdot \frac{{\partial p}}{{\partial y}}} \right). \hfill \\ \end{gathered}$$