150 Transactions. — Miscellaneous, 



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 produc- 

 tive 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 

 fi'om 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, be infinitesimals along 

 ax in the direction of the planet's course ; let aa', bV , 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, be 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 2^' represent the orbit with perihelion distance increased 

 beyond that of ^j, this latter representing the orbit if the sun were not to 

 expand into a nebula. Let the dotted cu'cle c represent the limits to which 

 the nebula has expanded when the planet passes aphelion. As the jolanet 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 jy", 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 dis- 

 tance. 



