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

Appendix I.

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

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

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