In compliance with the wishes of several members, I have inserted in this paper the solutions of the dynamical problems involved, whose truth I had before assumed.

The agency of lessened attraction as affecting any one planet, applies only to the period which elapses while the central mass is expanding to a nebula, and it will appear that the first revolution will especially be productive of altered eccentricity on this count. The following shows the action of these forces reduced to geometrical problems:—

Problem 1. Suppose a planet to be at that part of its orbit most distant from the sun, and, while in this position, suppose the mass of the sun suddenly diminished to a given extent,—required to trace the effect of this diminution of the sun's mass upon the orbit of the planet.

At present let the sun's mass be considered constant. Let the line ax fig. 1 be tangent to the curve at aphelion, and aa, ab, bc infinitesimals along ax in the direction of the planet's course; let aa', bb', cc', be infinitesimals representing the fall of the planet during the times contained respectively in aa, ab, ac, then aa' b' c' will be the path of the planet.

Now suppose the mass of the sun to be decreased, the infinitesimals aa, ab, bc will remain unaltered, but aa', bb', cc', etc., will each be diminished to a” b” c”. Then the curve aa” b” c” represents the new orbit. It falls without the old orbit, except at a where it coincides with it. Perihelion distance is therefore increased, as represented in fig. 2, by virtue of diminished attraction.

The amount of the lessening of the attractive force will depend upon the quantity of the sun's matter which expands beyond aphelion distance. The portion which so expands ceases to affect the path of the planet. As this increases the orbit will assume variously the forms of the ellipse, circle, ellipse (the foci being reversed), parabola and hyperbola. If the attraction towards the centre entirely ceased, the path would coincide with the line aa. These orbits are respectively shown in fig. 2.

In fig. 3 let p′ represent the orbit with perihelion distance increased beyond that of p, this latter representing the orbit if the sun were not to expand into a nebula. Let the dotted circle c represent the limits to which the nebula has expanded when the planet passes aphelion. As the planet is entirely in the nebula it will be subject to constantly and rapidly diminishing attraction as it approaches the centre, s, hence it will not pass along p’, but will move more slowly inwards (in agreement with the first problem), and will pass along the second dotted line p′, which shows great increase in perihelion distance.

The two actions which have now been discussed scarcely affect aphelion distance, but render the orbit more circular by increasing perihelion distance.

I have now to notice gaseous resistance and interchange of molecules, whose action will be found chiefly to diminish aphelion distance. The following 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 represents 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 a′ β′ γ′ are points in the new orbit.

This curve lies entirely within the other. Thus, by a retardation at perihelion, aphelion distance is diminished, 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-