Page:Philosophical Transactions of the Royal Society A - Volume 184.djvu/499

500

Determination of the "Null Point" and an alternative Method of calculating the Results.

As stated in the introduction we append a full analysis of the "null point" i.e., the point at which the radiation is self-eliminated.

Using a similar notation to that on p. 478, we have as the equation of condition

$\mathrm{M}\frac{d\theta}{dt} = a - \rho\theta\quad.\quad.\quad.\quad.\quad.\quad.\quad.\quad.\quad.\quad.\quad (1).$

where the temperatures are measured from that of the surrounding envelope.

For the sake of simplicity we can assume that the values of $$a$$ and $$\mathrm{M}$$ remain constant.

Integrating and putting $$\lambda = \rho/\mathrm{M}, \mu = a/\mathrm{M}$$, and determining the constant from the fact that when $$t = 0, \theta = -\,\theta_0$$, we obtain the equation

$e^{\lambda t} = \frac{\frac{\mu}{\lambda} + \theta_0}{\frac{\mu}{\lambda} - \theta}\,.\quad.\quad.\quad.\quad.\quad.\quad.\quad.\quad.\quad.\quad.\quad.\quad (2).$

If there had been no radiation, $$\rho = 0$$, and the equation of condition would have been $$d\theta/dt = \mu$$.

Integrating, and using the same constant as before

$\theta = \mu t - \theta_0\,.\quad.\quad.\quad.\quad.\quad.\quad.\quad.\quad.\quad.\quad.\quad (3).$

If we find the points of intersection of (2) and (3) one point $$\left(-\,\theta_0,\,0\right)$$ is that at which the experiment commenced, the other $$\left(\Theta,\,\mathrm{T}\right)$$ is the point on (2) at which the radiation is eliminated. It is more convenient, for experimental work, to obtain an expression involving $$\mathrm{T}$$ rather than $$\Theta$$; substituting therefore the value of $$\theta$$ given by (3) in (2), we obtain

$e^{\lambda\mathrm{T}} - \frac{\frac{\mu}{\lambda} + \theta_0}{\mu}\frac{e^{\lambda\mathrm{T}} - 1}{\mathrm{T}} = 0\,.\quad.\quad.\quad.\quad.\quad.\quad.\quad.\quad.\quad (4).$|undefined

This equation can be solved for $$\mathrm{T}$$ when the values of $$\lambda,\,\mu,\,\theta_0$$ are known.

In order to obtain $$\lambda,\,\mu$$, we can take the following observations: Commence an experiment at $$-\,\theta_0$$, note the time $$t$$ when $$\theta = 0$$, and again note the time $$t_2$$, when $$\theta = +\,\theta_0$$.

Then equation (2) gives the relations