= Solution
The dimensions are $[\rho]=ML^{-3}$, $[\sigma]=MT^{-2}$ and $[\lambda]=L$. Equating exponents in $LT^{-1}=M^{\alpha+\beta}L^{-3\alpha+\gamma}T^{-2\beta}$ yields
$$
\boxed{\alpha=-\frac12,\qquad\beta=\frac12,\qquad\gamma=-\frac12.}
$$
Thus the model has $v_{\rm ph}\propto\sqrt{\sigma/(\rho\lambda)}$. Its common dimensionless factor cancels in the ratio:
$$
\boxed{\frac{v_{\rm water}}{v_{\rm mercury}}=\sqrt{\frac{0.073}{0.490}\frac{13600}{1000}\frac{5}{10}}=1.0065\simeq1.01.}
$$
They are nearly equal within this capillary-wave model. The assumed <power law> omits gravitational restoring <forces> and finite-depth effects; <dimensional analysis> cannot establish those omissions for an arbitrary real surface <wave>.
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