A light source, which emits two wavelengths $\lambda_1=400 \mathrm{~nm}$ and $\lambda_2=600 \mathrm{~nm}$, is used in a Young's double slit experiment. If recorded fringe widths for $\lambda_1$ and $\lambda_2$ are $\beta_1$ and $\beta_2$ and the number of fringes for them within a distance $y$ on one side of the central maximum are $m_1$ and $m_2$, respectively, then