MEMOIR 



THE SECULAR VARIATIONS OF THE ELEMENTS OF THE ORBITS OF THE 



EIGHT PRINCIPAL PLANETS, 



MERCURY VENUS THE EARTH MARS, JUPITER, SATURN, URANUS, AND NEPTUNE. 



CHAPTER I. 

 ON THE SECULAR VARIATIONS OF THE ECCENTRICITIES AND PERIHELIA. 



1. We shall assume as the basis of our computation the following differential 

 equations, which determine the instantaneous variations of the eccentricities and 

 places of the perihelia of the planetary orbits at any time. These equations are 

 demonstrated by LA PLACE, in Book II, Chapter VII, of the Mecanique Celeste; 

 and by PONTECOULANT, in Book II, Chapter VIII, of his Theorie Analytique du 

 Syst&me du Monde, and are as follows : 



f t = | (o,o+(o, 8 )+(o,8)+&c. } i -E] 



* - 



di~ 



dhf_ I 

 dt = \ 



=-{(i,o)+<i,0+0,)+&e.}A'- 



&c. 



If we denote by e, e 1 , e", &c., js, rf, -a", &c., the eccentricities and longitudes of 



the perihelia of the orbits of Mercury, Venus, the Earth, &c., we shall have the 



following equations for the determination of these quantities : 



h=e sin or, h'=e l sin c/, K'=e" sin 



I =e cos or, Z' =e' cos or*, Z" =e" cos GT", 



Whence we deduce 



Ogn?" [oTIJZ"' &c.; 



rjl"\l'" &c.; 



, (A) 



r, &c., ) 

 ", &c. j 



n 



, &c.; tan =y tan CT '=, &c. (2) 



If h, Z, A', I', &c., are determined by the integrals of equations (A), for any time 

 whatever, and substituted in equations (2), we shall obtain the corresponding values 

 of e, e", &c., cr, cr 7 , &c. 



May, 1871. \ ( 1 ) 



