﻿338 Prof. P. Lowell on the Asteroids. 



the integrals in the problem of three bodies cannot be 

 rigorously solved, and have to be expressed in series, so that 

 the result obtained depends upon the number of terms in- 

 cluded. Secondly, Poincare has latterly shown that the 

 series used by astronomers become eventually divergent so 

 that in time they cease to represent the action at all. 

 Finally, whereas Laplace and Lagrange thought they had 

 established the Eventual stability of the solar system, by 

 proving that so far as terms to the first order in the masses 

 are concerned there was no secular variation of the major 

 axes, it has since been demonstrated that in those of the 

 second order in the masses terms of the form £ cos and 

 t sin arise, called from their discoverer Poisson terms, and 

 by Haretu that in the third order terms in which t is ex- 

 plicitly represented make their appearance. So that we can 

 no longer assert the secular non-variability of the major 

 axes. 



These two points : that it is never safe to argue from a 

 single term, and that we can no longer assume the major 

 axes to be in the long run invariable, entirely vitiate deduc- 

 tions too hastily drawn from the pendulum effect discovered 

 by Laplace. Furthermore, it appears that the received 

 criterion for libration does not hold throughout the action. 

 This will be seen when we examine the subject analytically. 



Tisserand, in his excellent expose in his Meeanique Celeste 

 and elsewhere, discusses only the case where Jupiter's orbit 

 is supposed circular. (Jallandreau does the same. Further- 

 more, Tisserand neglects the term ~j- as being negligible, 

 which turns out not to be the case. 



2. We shall now investigate the subject with more parti- 

 cularity. In so doing we shall find that libration does not 

 account for the gaps in the asteroid belt nor in the rings of 

 Saturn, nor, finally, for the entire lack of commensurability 

 in the periods of all the major planets. 



"We will begin by consideration of the case of asteroids 

 travelling at a mean distance from the Sun such as to make 

 their mean motions nearly twice that of Jupiter, those, there- 

 fore, in the neighbourhood of the gap n — 2n'. We shall not 

 assume Jupiter's orbit circular, but shall take the ellipticity 

 of that orbit into account, designating everything relating 

 to Jupiter by a dash ; the elements of the asteroid being 

 unaccented. 



The terms of the perturbing function which are important 

 in the case are those dependent on the argument / — 21'. We 



