In a normal range of temperatures, the resistance of a metallic conductor increases linearly with the rise in temperature. Therefore, the resistance-temperature graph is a straight line as shown in Fig.

In the above image, R0 starts from a certain resistance value because every material has some resistance, and as temperature increases, the resistance increases, respectively.

Consider a metallic conductor having resistance R_oat 0^oC and R₁ at t1 ^oC . In the normal range of temperatures, the increases in resistance are (i.e., R1 – R0).

(i) Directly proportional to the initial resistance i.e.,

R₁-R₀ ∝ R₀

(ii) Directly proportional to the rise in temperature i.e.,

R₁-R₀ ∝ t₁

By combining the two equations above, we get.

R₁-R₀ ∝ R₀t₁ ----------(1)

Where α₀ is a constant and is called temperature co-efficient of resistance at 0 degree celcius and the value depend on the nature of material and temperature

From above eq (1) we can write as R₁=R₀(1+ α₀t₁) -------(2)

And ,α₀ = (R₁-R₀)/(R₀ x t₁)

Temperature coefficient α₁ at t₁⁰c is given by

α₀R₀= α₁R₁

α ₁= α₀R₀/ R₁

α₁= α₀R₀/R₀(1+ α₀t1) ------------- from eq(2)

α₁ = α₀/(1+α₀t₁)

similarly for α₂ = $\frac{{\alpha }_0}{1+{\alpha }_0{t}_2}$