1. **Problem Statement:** We are given the equation $$R \sin \beta = P \sin \gamma + Q \sin \alpha$$ and the substitutions $$\sin \beta = \frac{r}{om}, \quad \sin \gamma = \frac{p}{om}, \quad \sin \alpha = \frac{q}{om}$$. 2. **Substitution:** Substitute the sine values into the original equation: $$R \left(\frac{r}{om}\right) = P \left(\frac{p}{om}\right) + Q \left(\frac{q}{om}\right)$$ 3. **Simplification:** Since all terms share the denominator $om$, multiply both sides by $om$ to eliminate it: $$Rr = Pp + Qq$$ 4. **Interpretation:** This equation relates the magnitudes of vectors $\overrightarrow{R}$, $\overrightarrow{P}$, and $\overrightarrow{Q}$ scaled by the lengths $r$, $p$, and $q$ respectively. 5. **Geometric Context:** The triangle with vertices $O$, $m$, and the third vertex shows vectors $\overrightarrow{Q}$, $\overrightarrow{R}$, and $\overrightarrow{P}$ originating from $O$. The angles $\alpha$, $\beta$, and $\gamma$ correspond to the angles between these vectors and sides of the triangle. 6. **Conclusion:** The equation $Rr = Pp + Qq$ is a vector relation derived from the sine rule and the given substitutions, expressing a balance between the components of the vectors scaled by the triangle's side lengths. This explanation helps understand how the original trigonometric relation transforms into a linear relation involving vector magnitudes and side lengths.