Page:Philosophical Transactions of the Royal Society A - Volume 184.djvu/2

to criticism—one being, that if it were absolutely applicable, the light of the sky should probably exhibit greater polarisation in a direction perpendicular to the sun's rays than it does. There is reason for believing, however, that the higher the station at which such observations are made the more complete is the polarisation. In any case, before this can be considered a valid objection, we have to know more than we do at present regarding the condition of the light reflected back from the earth and from the particles themselves.

It must also be admitted that grosser particles exist on some days, and that for these the loss of light must be in the form of $\mathrm {I} '=\mathrm {I} \epsilon ^{-\mu x}$. They would prevent a certain portion of the total light from reaching the observer, whilst the smaller one would selectively reflect, and hence the law would not hold absolutely good, but the small number of gross particles, compared with the fine ones, would not appreciably alter the formula used except the coefficient $\kappa$.

Another objection which has been advanced by Mr. S. P. Langley in a private letter against the adoption of the formula is, that if the spectrum were observed with a large dispersion, it will be seen that as the altitude of the sun diminishes the atmo spheric lines increase in intensity, and that these must obey the laws of ordinary absorption. This is an objection which at first sight may seem fatal, not to the correctness of the observations, but to the adoption of the law above quoted; but it must be remembered that these special absorptions occupy a very limited area com pared with the rest of the spectrum, and that they would practically disappear when the whole loss of light is under consideration, more especially as they would themselves obey the ordinary law of absorption. At any rate, the formula adopted appears to suit the case, and it must be borne in mind, that the results obtained by the inte gration of the spectrum luminosities bear out the formula which has been universally adopted by astronomers as representing the corrections to be made in star magnitudes when the stars are observed at different altitudes above the horizon.

XXIV.—The Integration of the Spectrum Luminosities at different Solar Altitudes
equivalent to the Luminosity of Monochromatic Light at the same Altitudes.

In Section XVIII. of the paper of which this is a continuation it was shown that the areas of the curves obtained from the formula $\mathrm {I} '=\mathrm {I} \epsilon ^{-\kappa x\lambda ^{-4}}$ being capable of being represented by $\mathrm {I} '=\mathrm {I} \epsilon ^{-\mu x}$ an important deduction could be made.

For the area of the curve is $\epsilon ^{\kappa x}\left(a\epsilon ^{\lambda ,^{-4}}+b\epsilon ^{\lambda ,,^{-4}}+c\epsilon ^{\lambda ,,,^{-4}}+\&\mathrm {c} .\right).$ $a$, $b$, $c$, $\&\mathrm {c} .$, being the original luminosities of the different rays, must then be represented by $\mathrm {I} =\mathrm {I} \epsilon ^{-\kappa x\Lambda ^{-4}}$ where $\Lambda$ is some one ray, that is $\mu =\kappa \Lambda ^{-4}$ or ${\frac {\mu }{\kappa }}=\Lambda ^{-4}$; ${\frac {\mu }{\kappa }}$ is a constant and was found from the observations to be $105$ (on the scale used for $\Lambda ^{-4}$), which is equivalent to $\lambda \;5770$, or a ray near the place of maximum luminosity. Hence the visual observations of total sunlight at different zenith distances are equivalent to observing the alteration in intensity of one single ray of that wave length in its spectrum,^[1] and,

1. This, of course, is only true within certain limits, but it holds for any thickness of atmosphere through which observations were taken.