I have now to notice gaseous resistance and interchange of molecules, whose action will be found chiefly to diminish aphelion distance. The followmg problem demonstrates decrease of aphelion distance by a resistance at perihelion.
Problem 2. Suppose a planet to be at that part of its orbit nearest to the sun, and, when in that position, suppose a retarding force to act upon it,—required to trace the effect of this upon the orbit of the planet.
Let Px represent a tangent to perihelion, and pa, ab, bc be components in direction pn, passed over in three successive infinitesimals of time. Let a α, b β, c γ represent the total fall towards the sun in the same intervals. Then p α β γ, represents the orbit. Now let the velocity in the direction px be diminished by the retarding force, and let the spaces pa′, a′b′, b′c′ represent the components in the direction px in the same infinitesimals of time. The components towards the sun remaining the same draw αα′ ββ′ γγ′ parallel to px, then α′ β′ γ′ are points in the new orbit.
This curve lies entirely within the other. Thus, by a retardation at perihelion, aphelion distance is dimmished, as shown in fig. 5. If this retardation is great enough, the orbit may become a circle or an ellipse with foci reversed, as shown in fig. 5. The general action of gaseous resistance is to convert the energy of the system into heat by gradually drawing the planet into the sun, or to the centre of attraction. It is maximum at perihelion, for there the density of the nebula is greater than at any other part of the orbit. Molecular exchange results from the varying densities of the different parts of the system. The planets are cooler than the central parts of the nebula, and will most likely be denser than the matter surrounding them in their path, and have sufficient attractive power to collect the heavy molecules in their vicinity. The temperature of the surface of the planet will be raised to an unknown extent by its immersion in the nebula and its progress towards perihelion. Its light molecules have their velocity so increased as to escape the planet, while the heavier molecules of the vicinity, with their lower velocity (though equal temperature), will be attracted, picked up, and become permanently part of the planet. A greater proportion of heavy molecules will be found towards perihelion, for at the centre of the nebula will probably be its greatest density, and the original expansion of the central mass into a nebula will result in the more rapid outward escape of the light molecules compared with the heavy, in obedience to the laws of gaseous diffusion. Thus the accretion of molecules to the planet will be maximum at perihelion distance. Its effect will be to retard the motion of the planet, as, in order to give its own velocity to a molecule, it will impart some of its energy. The escape of the light molecules will not affect the planet's orbit. We find therefore that gaseous re-
