Page:Philosophical Transactions of the Royal Society A - Volume 184.djvu/22

$\mu ^{\prime }$ and $\mu$ are the exponential coefficients as before, $\alpha ^{\prime }$ and $\alpha$ are the constants in the formula, $\mathrm {I} \alpha ^{z}=\mathrm {I} ^{\prime }$, $z$ being the air thickness, and $\mathrm {I}$ and $\mathrm {I} ^{\prime }$ the intensities before and after transmission.

We are now in a position to determine the constant $\kappa$ in the formula

$\mathrm {I} ^{\prime }=\mathrm {I} \epsilon ^{-\kappa x\lambda ^{-2}}$

at sea-level with the barometer at 30 in. and in a very clear sky, for as

$\kappa \Lambda ^{-4}=\mu _{1}$,

therefore

$\kappa ={\frac {\mu ^{\prime }}{\Lambda ^{-4}}}-{\frac {.453}{308}}=.00146$;

in the case of fairly clear skies,

$\kappa ={\frac {.497}{308}}=.00161$.

In Part I. of this paper the minimum value of $\kappa$ was found to be .0013, and a mean value about .0017, so that these observations are fairly accordant.

$\kappa$ may be taken to be a measure of the number of particles the rays encounter, and thence it may be concluded that the number of particles at any thin layer of the atmosphere is $\alpha h$. The formula, therefore, for the scattering of a ray of any wave-length at any altitude becomes

$\mathrm {I} ^{\prime }=\mathrm {I} \epsilon ^{-ch^{2}x\lambda ^{-}4}$

where $c$ is a constant, $h$ the height of the barometer, and $x$ the air thickness, those of the zenith being "unity."

In comparing the comparative scattering of a ray at the same zenith distance, but at different altitudes, $\lambda \alpha x$ are constants, and

$\mathrm {I} ^{\prime }=\mathrm {I} \epsilon ^{-mh^{2}}$

where $m$ is a constant and $h$ the variable. This formula and that of the law of error are identical.

XXXV.—Conclusions.

In conclusion, it should be remarked that the loss of light as light from transmission through the atmosphere is, and must be, very different to that of the heating effect of the solar radiation. The latter is not principally dependent on scattering by small particles, but on the absorption of aqueous vapour, which is a very different matter. Langley has shown that the heating effect diminishes much more rapidly as the barometric pressure is diminished than is usually supposed, and this is not to