EEMARKS ON THE SMALL PLANETS. 209 



between 0° and 360^. This, astronomers had already remarked, 

 when as yet they \vere acquainted with but from 12 to 15 asteroids, 

 and it has been confirmed liy further discoveries : it is a fact highly 

 worthy of attention, on account of the consequences which may per- 

 haps be some day derived from it in relation to the origin and forma- 

 tion of the planets. Unfortunately it does not seem easy to detect 

 the law of the phenomenon ; but in this there is nothing surprising. 

 For, on one hand, the perihelia and nodes are constantly being dis- 

 placed, and, on the other, the plane of the ecliptic, itself movable, 

 is unimportant in the general aggregate of the solar system. It was 

 natural, then, to inquire whether we might not arrive at a more marked 

 convergence, by substituting for the lines of the nodes of the plane- 

 tary orbits the intersections of those orbits with a fundamental plane, 

 such as the invariable plane of the solar system, or even the equator 

 of the sun itself. "At the epoch when MM. Midler, Cooper, d' Ar- 

 rest, &c., undertook these researches and others of the same kind, 

 their investigations embraced scarcely half the planets now known, 

 nor did they, moreover, pursue a uniform method. By M. Burat and 

 myself all the calculations of these astronomers have been repeated, 

 adopting the numbers which appeared to ns most worthy of confi- 

 dence, as well for the elements of the asteroids as for the determina- 

 tion of the solar equator or invariable plane. We have, besides, con- 

 formed to a constant rule in our search for the centre of convergence of 

 a system of lines of the same kind. This rule, at once simple and 

 natural, consists in regarding these lines as so many equal forces and 

 in seeking the direction of their resultant. The same calculation 

 gives also the quantity of this resultant, a quantity evidently so much 

 the more considerable as the convergence is more marked ; for this 

 reason we give the name of coe^icient of convergence to the quotient of 

 the total resultant by the sum of the components. 



Convergence of the perihelia. — The rule just indicated gives us 47^ 

 20' for the longitude of the perihelia on the celestial sphere, and 

 0,255 for a coefficient of convergence. The hemisphere, which has 

 for its pole the point thus determined, comprehends the perihelia of 

 46 asteroids, having a mean eccentricity marked byO, 1G8. The 

 opposite hemisphere contains but 24 perihelia, and the mean eccen- 

 tricity of the corresponding orbits is but 0,137. The centre of con- 

 vergence determined by M. Ma^dler for the first 58 asteroids had for 

 its longitude 52° 25', and approaches, consequently, much more 

 than ours the group of the Pleiades, where, as is known, M. Maedler 

 places the centre of the immense orbit described in space by the sun 

 followed by its retinue of planets. 



Convergence of ascending nodes. — The point of convergence of the 

 70 ascending nodes, on the plane of the ecliptic, has for its longitude 

 135° 53', and the coefficient of convergence is 0,256, very nearly as 

 for the perihelia. Forty-six ascending nodes are found in the hem- 

 isphere which has for its pole the <;entre of convergence, and 24 in 

 the opposite hemisphere. The ratio of these two numbers is the 

 same as for the perihelia, and we M-ill recall, in this connexion, a 

 singular remark which .was made by M. Cooper some years ago, that 

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