ABSTRACTS 329 



Asymptotic Planetoids 

 By Professor Daniel Buchanan, M.A., Ph.D., F.R.S.C. 



(Read May Meeting, 1922) 

 {Abstracty 



Near the two vertices of the equilateral triangles which may be 

 described on the line joining the sun and Jupiter, as base and in the 

 plane of Jupiter's orbit, are to be found six planetoids, four at one 

 vertex and two at the other. These six planetoids are in the vicinity 

 of two of the five well-known points of libration in the problem of three 

 bodies. The other three points are collinear and lie on the line joining 

 the sun and Jupiter. One point lies between the sun and Jupiter, 

 another on the side of the sun remote from Jupiter, and the third on 

 the side of Jupiter remote from the sun. In the case of the sun and 

 the earth, one of these straight line equilibrium points, viz., the one 

 on the side of the earth remote from the sun, has a physical significance 

 in that it may account for the Gegenschein. The equilateral triangle 

 points of libration were first considered by Lagrange, in his celebrated 

 prize memoir of 1772, as "pure curiosities," but recent astronomical 

 discovery has shown that these points likewise have some physical 

 significance attached to them. 



The problem considered in this paper is to determine orbits for 

 the above mentioned planetoids, which will be asymptotic to the 

 equilateral triangule points of libration, that is, orbits which will 

 approach these points as the time approaches infinity. We have 

 designated planetoids which move in such orbits as "Asymptotic 

 Planetoids." The masses of these planetoids are not considered to 

 be infinitesimal, and their perturbations upon the sun and Jupiter are 

 determined. The orbits which have been found are of spiral type. 

 They are two-dimensional and lie in the plane of Jupiter's orbit. 



The differential equations of motion are integrated, according to 

 the methods of Poincaré,^ as power series in terms of the type e'^'P (/), 

 where c is a constant whose real part is different from zero, and P{t) 

 is a periodic function of /, or, in particular, a constant. Suppose 

 c = a-f V — 1 b. Then the orbits will approach the equilibrium points 



^The paper will appear in an early issue of the Transactions of the American 

 Mathematical Society. 



''Les Methods Nouvelles de la Mécanique Céleste, Vol. 1. 



