At certain temperature, \(\operatorname{rod} \mathrm{A}\) and \(\operatorname{rod} \mathrm{B}\) of different materials have lengths \(L_A\) and \(L_B\) respectively: Their coefficients of linear expansion are \(\alpha_A\) and \(\alpha_B\) respectively. It is observed that the difference between their length's remains constant at all temperatures. The ratio \(\mathrm{L}_{\mathrm{A}}: \mathrm{L}_{\mathrm{B}}\) is given by

Length of \(\operatorname{rod} \mathrm{A}\) at temperature \(\mathrm{t}=\mathrm{L}_{\mathrm{A}}+l_{\mathrm{A}} \alpha_{\mathrm{A}} \Delta \mathrm{t}\)
Length of \(\operatorname{rod} \mathrm{B}\) at temperature \(\mathrm{t}=\mathrm{L}_{\mathrm{B}}+l_{\mathrm{B}} \alpha_{\mathrm{B}} \Delta \mathrm{t}\)
Length of \(\operatorname{rod} A-\) length of \(\operatorname{rod} B\)
\(=\left(\mathrm{L}_{\mathrm{A}}-\mathrm{L}_{\mathrm{B}}\right)+\left(l_{\mathrm{A}} \alpha_{\mathrm{A}}-l_{\mathrm{B}} \alpha_{\mathrm{B}}\right) \Delta \mathrm{t}\)
For difference in the length to be constant, coefficient of \(\Delta t\) must be zero.
\(\begin{array}{ll}
\therefore & \quad l_{\mathrm{A}} \alpha_{\mathrm{A}}-l_{\mathrm{B}} \alpha_{\mathrm{B}}=0 \\
\therefore & l_{\mathrm{A}} \alpha_{\mathrm{A}}=l_{\mathrm{B}} \alpha_{\mathrm{B}} \\
\therefore & \frac{l_{\mathrm{A}}}{l_{\mathrm{B}}}=\frac{\alpha_{\mathrm{B}}}{\alpha_{\mathrm{A}}}
\end{array}\)