Page:On the Influence of Carbonic Acid in the Air upon the Temperature of the Ground.pdf/24

(without alteration of their other properties, as height, compactness, &c.), the ground will undergo the same variations of temperature. Now it will be shown in the sequel that a variation of the carbonic acid of the atmosphere in the same proportion produces nearly the same thermal effect independently of its absolute magnitude (see p. 265). Therefore we may calculate the temperature-variation in this case as if the clouds covered the ground with a thin film of the albedo $0.78$ ($\nu =0.22$, see p. 256). As now on the average ${\text{K}}=1$ and ${\text{W}}=1$ nearly, and in this case $\beta$ is about $0.79$, the effect on the clouded part will be only $0.25$ of the effect on parts that have $\nu =1$. If a like correction is introduced for the ocean ($\nu =0.925$) on the supposition that the unclouded part of the earth consists of as much water as of solid ground (which is approximately true, for the clouds are by preference stored up over the ocean), we find a mean effect. of, in round numbers, 60 p. c. of that which would exist if the whole earth's surface had $\nu =1$. The snow-covered parts are not considered, for, on the one hand, these parts are mostly clouded to about 65 p. c.; further, they constitute only a very small part of the earth (for the whole year on the average only about 4 p. c.), so that the correction for this case would not exceed 0.5 p. c. in the last number 60. And further, on the border countries between snowfields and free soil secondary effects come into play (see p. 257) which compensate, and perhaps overcome, the moderating effect of the snow.

In the foregoing we have supposed that the air is to be regarded as an envelope of perfectly uniform temperature. This is of course not true, and we now proceed to an examination of the probable corrections that must be introduced for eliminating the errors caused by this inexactness. It is evident that the parts of the air which radiate to space are chiefly the external ones, and on the other hand the layers of air which absorb the greatest part of the earth's radiation do not lie very high. From this cause both the radiation from air to space ($\beta \gamma \theta ^{4}$ in eq. 1) and also the radiation of the earth to the air ($\beta \gamma \nu ({\text{T}}^{4}-\theta ^{4})$ in eq. 2), are greatly reduced, and the air has a much greater effect as protecting against the loss of heat to space than is assumed in these equations, and consequently also in eq. (3). If we knew the difference of temperature between the two layers of the air that radiate to space and absorb the earth's radiation, it would be easy to introduce the necessary correction in formulæ (1), (2), and (3). For this purpose I have adduced the following consideration.

As at the mean composition of the atmosphere (${\text{K}}=1$,