French National Institute. ?05 



Mathematics. 



* c To give a theory of the perturbations of the planet Pallas, 

 discovered by Dr. Olbers." 



Geometricians have given, with sufficient exactness and 

 extent, the theory of the perturbations of all the old planets, 

 and of those which may be still discovered if confined within 

 the same zodiac, and if they have only a very small eccentri- 

 city . Mercury till the present time was considered as the most 

 eccentric of all the planets, and at the same time that which 

 had the greatest inclination ; but the smallness of its mass, 

 and its position at one of the limits of the planetary system, 

 render it very little calculated to produce very sensible alter- 

 ations in the motion of the other planets. Uranus, disco- 

 vered twepty years ago by Dr. Herschel, is placed at the 

 other limit of the system, with a small mass and little ec- 

 centricity : jt has also the smallest of all the inclinations 

 known ; so tfiat the formulae employed for Jupiter and Sa- 

 turn have been more than sufficient for that new planet. 

 Ceres, discovered a few years ago by M. Piazzi, having a 

 considerable eccentricity and an inclination of \0° 38', must 

 be subject to strong and numerous inequalities. It, how- 

 ever, appears that all the astronomers who have endeavoured 

 to determine them have been contented with known for- 

 mulae, the development of which does not exceed the pro- 

 ducts of the three dimensions of the inclinations and eccen- 

 tricities. Those of five dimensions have been employed in 

 the Mecanique Celeste for a particular case, according to 

 a formula of M. Burckhardt, The same astronomer has 

 since presented to the National Institute the general and 

 complete development of the third, fourth, and fifth orders; 

 but this degree of precision would certainly not be sufficient 

 for the planet Pallas, whose eccentricity is even greater than 

 that of Mercury, and inclination 34° 37'; that is to say, 

 five times greater than that of any other known planet* ft 

 is even difficult to conjecture what will be the powers anfi 

 the dimensions of the products which it may be allowable 

 to neglect ; and the calculations may be of siich length, and 

 ^he formulae so complex., that they might frighten astrono- 

 mers the best qualified to execute them. This consideration 

 has induced the Class of the Mathematical and Philoso- 

 phical Sciences to propose this subject as the question for 

 which it will adjudge a prize on the first Monday of Mes- 

 sidor, year 14. It therefore reauests geometricians and 

 £stroiiomers to discuss completely all the inequalities of this 



theory^ 



