636 Mr W. HOPKINS, ON THE EXTERNAL TEMPERATURE OF THE EARTH, 



its temperature at the planet's surface, that it may be enabled to transmit a given quantity 

 of heat in a given time. 



The effect of solar radiation is the same as if, instead of proceeding from a single focus, 

 it proceeded generally from surrounding space, the intensity of radiation on any point of 

 the Earth's surface, omitting daily variations, being a function of the latitude of the place 

 and the time of the year. It will be my object, in the latter part of this paper, to examine 

 the data which we possess for determining numerically what would be the effect of an 

 increase or diminution of our atmosphere on terrestrial temperature, and also what is the 

 actual effect of the sun's heat on that temperature. We shall then be able to judge what 

 may be the probable influence of similar causes on the temperature of the other planets. 

 But before I proceed to these points, I shall make some remarks on the mathematical 

 treatment of the problem which has for its object the determination of the actual pressure, 

 temperature, and density of the Earth's atmosphere. 



12. Conceive a cone of which the angle at the apex is indefinitely small, the apex 

 coinciding with the center of the Earth. If this cone be produced through the Earth's 

 atmosphere, the same quantity of heat must pass upwards in the same time through every 

 transverse section of the cone comprised within the atmosphere, the temperature being con- 

 sidered as steady. Let q denote the quantity of heat thus transmitted in a unit of time 

 through any section at the height x above the Earth's surface; then must we have 



g-o (0. 



ax 

 If the transmission take place by conduction, we have 



q = k.-f, 

 ax 



where £ is the temperature at the height r, and k the conductive power, probably some 

 function of the density p. If the transmission take place by radiation within certain finite 

 distances, or by convection, or by all three modes conjointly, it is manifest that q must 

 depend on the difference of temperatures at the point whose height is * and at surrounding 



rlY 



points, and the expression for it must therefore involve the differential coefficient — * , 



dx 



so that equation (l) will necessarily be a differential equation of the second order. Another 



equation must give us a relation between p, p, and x. If the atmosphere be at rest, this 



equation is 



dp 7 s 



d, = - g V^^P ( 2 )> 



where r = radius of the Earth. A third relation between p, p, and £ depends merely on 

 the property of air and not on the particular conditions under which it exists as an atmo- 

 sphere surrounding a particular planet. We have as the result of experiment, 



p = aV(l + a£) (3), 



