Page:Elementary Text-book of Physics (Anthony, 1897).djvu/399

§ 316] This equation may be put in the form used by Tait, if we write $$b = at_{n}$$ and $$c = -\frac{a}{2}\cdot$$ We then have The electromotive force in the circuit can become zero when either of these terms equals zero. It has been already stated that when $$t_{1} = t_{2},$$ or when both junctions are at the same temperature, there is no electromotive force in the circuit. When $$\tfrac{1}{2}(t_{1} + t_{2})$$ equals $$t_{n},$$ or when the mean of the temperatures of the hot and cold junctions equals a certain temperature, constant for each pair of metals, there will be also no electromotive force in the circuit. This temperature $$t_{n}$$ is that which has already been called the neutral temperature. The formula also assigns the value to that temperature $$t_{1}$$ at which, for fixed values of $$t_{n}$$ and $$t_{2},$$ the electromotive force in the circuit is a maximum. If we represent the difference between $$t_{n}$$ and $$t_{1}$$ by $$x,$$ then $$t_{1} = t_{n} \pm x.$$ Using this value in the formula, we obtain $$E_{1} - E_{2} = \frac{a}{2}((t_{n} - t_{2})^2 - x^2)$$ This is manifestly a maximum when $$x = 0.$$ The electromotive force in a circuit is then, according to the formula, a maximum when the temperature of one junction is the neutral temperature.

The formula also shows that the thermoelectric power is zero when $$t_{1} = t_{n}.$$ We may set $$E_{1} = A + at_{n}t_{1} - \frac{a}{2}t_{1}^2 .$$ Now if $$t_{1}$$ take any small increment $$\Delta t_{1}, \, E_{1}$$ has a corresponding increment $$\Delta E_{1}.$$ Hence we have $$E_{1} + \Delta E_{1} = A + at_{n}t_{1} - \frac{a}{2}t_{1}^2 + at_{n}\Delta t_{1} - at_{1} \Delta t_{1},$$ if we neglect the term containing $$\Delta t_{1}^2 .$$ From this equation we obtain $$\frac{\Delta E_{1}}{\Delta t_{1}} = at_{n} - at_{1},$$ which in the limit, as $$\Delta t_{1}$$ becomes indefinitely small, is the thermoelectric power at the temperature $$t_{1}.$$ It is positive for values of $$t_{1}$$ below $$t_{n};$$ is zero for $$t_{1} = t_{n},$$ and negative for higher values of $$t_{1}.$$ That is, if we assume $$t_{1} = t_{2}$$ lower than $$t_{n},$$ and then gradually raise the temperature $$t_{1},$$ the thermoelectric power at the heated junction is at first positive, but