SECULAR VARIATIONS OF THE PLANETARY ORBITS. 263 



but, being based upon vahies of the masses wliicli were, to a certain 

 extent, gratuitously assumed, it was desirable tliat the important truths 

 which it indicated should be established iudependeutl^' of any conside- 

 rations of a hypothetic character. This magnificent generalization was 

 elTected by La Place. He proved that, whatever be the relative masses 

 of the planets, the roots of the equations which determine the periods 

 of the secular inequalities will all be real and unequal, provided the 

 bodies of the system are subjected to this one condition, that they all 

 revolve round the sun in the same direction. This condition being satisfied 

 by all the members of the solar system, it follows that the orbits of the 

 planets will never be very eccentrical or much inclined to ea(;h other by 

 reason of their nnitual attraction. The important truths iii relation to 

 the forms and positions of the planetary orbits are embodied in the two 

 following theorems by the author of the Mecanique Celeste : I. If the mass 

 of each planet he multiplied hjj the product of the square of the eccentricity 

 and square root of the mean distance, the sum of all these products will 

 always retain the same magnitude. II. If the mass of each planet he mul- 

 tiplied hy the product of the square of the inclination of the orbit and the 

 square root of the mean distance, the S7im of these products will always 

 remain invariable. Now, these quantities being computed for a given 

 epoch, if their sum is found to be small, it follows from the preceding 

 theorems that they will always remain so; consequently the eccentri- 

 cities and inclinations cannot increase indefinitely, but will always be 

 confined within narrow limits. 



In order to calculate the limits of the variations of the elements with 

 precision, it is necessary to know the correct values of the masses of all 

 the planets. Unfortunately, this hnowledge has not yet been attained. 

 The masses of several of the planets are found to be considerably ditler- 

 ent from the values employed by La Grange in his investigations. 

 Besides, he only took into account the action of the six principal planets 

 which are within the orbit of Uranus. Consequently his solution afforded 

 only a first approximation to the limits of the secular variations of the 

 elements. 



The person who next undertook the computation of the secular ine- 

 qualities was Pontecoulant, who, about the year 1831, published the 

 third volume of his Theoric Analytique du Systeme du Monde. In this 

 work he has given the results of his solution of this intricate problem. 

 But the numerical values of the constants which he obtained are totally 

 erroneous on account of his failure to employ a sufficient number of 

 decimals in his computation. Our knowledge of the secular variations 

 of the planetary orbits was, therefore, not increased by his researches. 



In 1839 Le Verrier had completed his computation of the secular ine- 

 qualities of the seven principal planets. This mathematician has given a 

 new and accurate determination of the constants on which the amount of 

 the secular inequalities depend; and has also given the coefficients for 

 correcting the values of the constants for differential variations of the 



