The apparent coefficients of volume expansion of a & liquid filled...

Let V be the volume of the liquid and $\Delta \text{T}$~the rise in temperature. Since apparent expansion = true expansion - expansion of vessel

$∴ \text{V}{γ}_{1} \Delta \text{T} = \text{V}\gamma \Delta \text{T-V}\left(3{α}_{1}\right) \Delta \text{T}$

or, ${γ}_{1} = \gamma + 3 {α}_{1}$ ...(i)
(for vessel A)

and~${γ}_{2} = \gamma + 3 {α}_{2}$ ...(ii)
(for vessel B)

where $\gamma$~is the coefficient of real expansion of the liquid.
Subtracting Eq. (ii) from Eq. (i),

${γ}_{1} - {γ}_{2} = 3 \left({α}_{1} - {α}_{2}\right)<br /><br />or,${α}_{1} - {α}_{2} = \left({γ}_{1} - {γ}_{2}\right)$/ 3$

or, ${α}_{2} = \left[ \left({γ}_{2} - {γ}_{1}\right)/ 3\right] + {α}_{1}$