REMARKS ON THE SMALL PLANETS, 211 



For the large For the small 

 planets. planets. 



Arc of the solar equator, reckoned from the as- 

 cending node of that equator on the ecliptic 

 to the centre of convergence 184° 9' 1G4° 50' 



Coefficient of convergence •' 0,904 0,624 



These coefficients are, we see, much more considerable than those 

 found above. Moreover, of the 70 ascending nodes, 62 are situated 

 in the same hemisphere with the centre of convergence, and only 

 eight in the opposite hemisphere. But, notwithstanding the apparent 

 singularity of this result, a little reflection will suffice considerably 

 to diminish its importance, for it arises, in great measure, from the 

 fact that the planetary orbits are less inclined to one another than to 

 the equator of the sun. We perceive, in effect, w^ithout difficulty, 

 a priori^ that closely adjacent planes, conformably with lines closely 

 approximate, must intersect another plane, which forms with each 

 of them an angle of some degrees. The same considerations explain 

 why the convergence is more marked in regard to the' great than the 

 small planets. It must be admitted, then, that this phenomenon has 

 not the importance attributed to it by Cassini and some modern 

 astronomers; but the calculation, of which we have just given the 

 results, is not less useful on that account for determining, better than 

 the nodes on the ecliptic could do, the position in space of the mean 

 orbit of the asteroids, of which the masses should be considered 

 as equal. We may, with M. d' Arrest, regard as the plane of the 

 mean orbit the plane which cuts the solar ecliptic according to the 

 line of convergence, under an inclination equal to the mean of the 

 inclinations of all the asteroids; that is to say, under an angle of 9° 

 58'. No doubt the inclijiation thus determined is not absolute, and 

 would change slightly at each new discovery. But as we find results 

 but little different when the calculation is confined in succession to 

 the 30, 40, or 60 first known asteroids, we are justified in concluding 

 that these results are sufficiently approximate to that w^hich the whole 

 number of the small planets would give, if all were discovered. 



RELATION BETWEEN THE ECCENTRICITIES AND INCLINATIONS OF THE ORBITS. 



M. d' Arrest, while remarking that great inclination is not always 

 accompanied by great eccentricity, and that a consideration of the 

 separate orbits guides us to no relation between these two elements, 

 conceives, nevertheless, that if these orbits are suitably grouped in 

 the order of their inclinations to the solar equator, the mean of the 

 inclinations, for each group, would increase with the mean of the 

 eccentricities ; he has even arrived, for the first thirteen planets 

 grouped by fours and by threes, at the following empirical formula, 

 in which i expresses the mean inclination, and e the mean eccen- 

 tricity of each group : 



e = 0.0S51 + 0.0068 {. 



M. Littrow, w^ho has given this formula in iiis Astronomy, (1854,) 



