210 EEMARKS ON THE SMALL PLANETS. 



in considering the 10, 20, 30, 40, or 50 planets first known, we shall 

 always find, nearly to a Tinit, the same number of perihelia and of 

 ascending nodes in the respective hemispheres where these points 

 have their centres of convergence. 



The inequality in the distribution of the ascending nodes on the 

 celestial sphere is still more marked when we take the angular sub- 

 divisions lower than 90°. If we divide, for instance, the ecliptic into 

 six arcs of GO'-', reckoning from the fifth degree of longitude, we find: 



From 5° to 65° 10 ascending nodes. 



From 65° to 125° 12 do. 



From 125° to 185° 23 do. 



From 185° to 245° 10 do. 



From 245° to 305° 6 do. 



From 305° to 5° 9 do. 



The general table shows, moreover, independently of the predomi- 

 nance of nodes in one part of the heavens, a general disposition of 

 these nodes in partial groups, isolated one from the other; for instance, 

 towards the 8th degree of longitude, the 43d, the 68th, the 80th, &c. 



It may be asked, as we have already observed, whether a more 

 perspicuous law might not be obtained by substituting for the plane 

 of the ecliptic the invariable plane or the plane of the solar equator. 

 We have made this double calculation for each of the 70 planets, be- 

 ginning with a determination as exact as possible of the position of 

 the invariable plane — that is to say, by taking account, in conformity 

 with the remarks of Poinsot, of the areas which arise from the rota- 

 tion of the sun and the great planets. This plane, whose elements 

 differ little from those which Laplace had adopted, is only inclined 

 about 1° 41' to the ecliptic. It hence results that the nodes group 

 themselves on the two planes much after the same manner, and that 

 the consideration of the invariable plane casts no new light on the 

 question. 



At first view it would seem to be quite otherwise with regard to 

 the plane of the solar equator. As early as the beginning of the 

 last century, (Memoires de V Academie des Sciences^ 1734,) Cassini had 

 observed that the intersections of the equator of the sun with the 

 orbits of Venus, the earth. Mars, Jupiter, and Saturn, grouped them- 

 selves on an arc of less than 25 degrees in extent. In 1851 M. d' Ar- 

 rest made an analogous calculation for the first thirteen asteroids, 

 and found that, notwithstanding a more considerable mean divergence 

 of the ascending nodes on the solar equator, eleven of those nodes 

 out of thirteen were situated in the same hemisphere with the nodes 

 of the ancient planets. We have taken up and completed the work, 

 as well for the eight great planets now known as for the 70 asteroids, 

 and we give the results at which we have arrived, while adopting 

 for the position of the solar equator the numbers ascertained by M. 

 Langier : 



Longitude of the ascending node of the solar equator on the 



ecliptic 75° 8' 



Inclination • 7° 9' 



