REMARKS ON THE SMALL PLANETS. 217 



magnitude of the inclinations and eccentricities of the orbits. Hence, 

 as has been already said, there are several of these orbits of which 

 the elements are at present known with bnt very little exactness. A 

 more rigorous determination of these elements, and the investigation 

 of their secular variations, constitute one identical problem whose 

 solution exacts the employment of new methods. There have been 

 constructed, indeed, from year to year since the beginning of the 

 century, tables of the four older asteroids by following the processes 

 of interpolation indicated by Gauss and Encke. MM. Bruno w and 

 Hansen have even extended these processes to some of the more re- 

 cently discovered planets, such as Flora. But we arrive in this way 

 only with great difficulty at the analytic expression of the periodic 

 or secular inequalities. M. Hansen has recently published a new 

 theory of the planetary perturbations, the advantages of which are 

 pre-eminently decisive when the orbits are very much inclined or 

 very eccentric, and the distinguished astronomer has himself applied 

 this theory to the calculation of the perturbations of Egeria. M. Le 

 Yerrier, on his part, has discovered the great inequality of Pallas by 

 a method wholly different, to which M. Hoiiel has lately added some 

 improvements which facilitate its application. Such are very nearly 

 the attempts which have been made by geometers up to the present 

 time to master the theory of the small planets. 



We have here as well one of the most recent as most interesting 

 of astronomical questions. The perturbations, necessarily very great, 

 which result from the attraction exerted by Jupiter, will give, when 

 they shall have been calculated, a very exact determination of the 

 mass of that planet. It will be more difficult to calculate the indi- 

 vidual masses of the asteroids, but we shall arrive probably without 

 excessive difficulty at a knowledge of the precise value of their com- 

 bined masses. M. Le Verrier has already deduced from his investi- 

 gations respecting Mercury and Mars the higher limit of that value, 

 and it is to the following formula that he has himself reduced his re- 

 markable conclusions : 



1st. Besides the planets Mercury, Venus, the Earth, and Mars, 

 there exists between the Sun and Mercury a ring of asteroids which 

 collectively constitute a mass comparable to that of Mercury itself. 



2d. At the distance of the earth from the sun there is found a 

 second ring of asteroids whose mass is at most equal to the tenth part 

 of the mass of the earth. 



3d. The total mass of the asteroids comprised letween Blars and 

 Jvpiter is, at most, equal to the third of the mass of the eai^th. 



4th. The masses of the two latter groups are complementarj^ one 

 of the other. Ten times the mass of the group situated at the dis- 

 tance of the earth, plus three times the total mass of the small planets 

 situated between Mars and Jupiter, form a sum equal to the mass of 

 the earth. 



If we compare these results of the calculus with what has been 

 said above of the smallness of the volume of the asteroids, we shall 

 arrive at a remarkable consequence. By supposing that the mean 

 density of these bodies is equal with that of the earth, the united 



