At certain temperature, rod A and \(\operatorname{rod} \mathrm{B}\) of different materials have lengths \(L_A\) and \(L_B\) respectively. Their co-efficients of linear expansion are \(\alpha_A\) and \(\alpha_B\) respectively. It is observed that the difference between their lengths remain constant at all temperatures. The ratio \(L_A / L_B\) is given by

\(L_B^{\prime}=L_B\left(1+\alpha_B \Delta \theta\right)\) and \(L_A^{\prime}=L_A\left(1+\alpha_B \Delta \theta\right)\). So that
\(L_B^{\prime}-L_A^{\prime}=L_B-L_S+\left(L_B \alpha_B-L_A \alpha_A\right) \Delta \theta\)
So \(\left(L_B^{\prime}-L_A^{\prime}\right)\) will be equal to \(\left(L_B-L_A\right)\) at all temperature if, \(L_B \alpha_B-L_A \alpha_A=0\)
\([\operatorname{as} \Delta \theta \neq 0] \text { or } \frac{L_B}{L_A}=\frac{\alpha_{A}}{\alpha_B}\)
i.e., the difference in the lengths of the two rods will be independent of temperature if the lengths are in the inverse ratio of their coefficients of linear expansion.