Derivation of formula for temperature coefficient of resistance
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$$
\frac{R_{t_1}}{R_o} = \frac{t_o + t_1}{t_o + 0}, \tag{1}
$$
$$
\frac{R_{t_2}}{R_o} = \frac{t_o + t_2}{t_o + 0}, \tag{2}
$$
$$
(1) \div (2) \Rightarrow \frac{R_{t_1}}{R_{t_2}} = \frac{t_o + t_1}{t_o + t_2}
 = \frac{(t_o + t_2) + (t_1 - t_2)}{t_o + t_2} = 1 + \frac{t_1 - t_2}{t_o + t_2}
 = 1 + \alpha_{t_2} (t_1 - t_2),
$$
where \$\alpha_{t_2} = 1 / (t_o + t_2)\$, and
$$
  R_{t_1} = R_{t_2} + \alpha_{t_2} R_{t_2} (t_1 - t_2).
$$