Solution:

The coefficient of apparent expansion of mercury in a vessel is given by:

\[
\gamma_{\text{apparent}} = \gamma_{\text{mercury}} - \alpha_{\text{vessel}}
\]

For the glass vessel:
\[
153 \times 10^{-6} = \gamma_{\text{mercury}} - \alpha_{\text{glass}}
\]

For the steel vessel:
\[
144 \times 10^{-6} = \gamma_{\text{mercury}} - \alpha_{\text{steel}}
\]

Subtracting these two equations, we get:

\[
153 \times 10^{-6} - 144 \times 10^{-6} = \alpha_{\text{steel}} - \alpha_{\text{glass}}
\]

Substitute \(\alpha_{\text{steel}} = 12 \times 10^{-6}/^\circ \text{C}\):

\[
9 \times 10^{-6} = 12 \times 10^{-6} - \alpha_{\text{glass}}
\]

Solving for \(\alpha_{\text{glass}}\):
\[
\alpha_{\text{glass}} = 3 \times 10^{-6}/^\circ \text{C}
\]