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. 



(A) 



CHAPTER I. 



ON THE SECULAR VARIATIONS OF THE ECCENTRICITIES AND PERIHELIA. 



1. We shall assume as the basis of otir 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 Tlieorie Analytique du 

 Systeme du Monde, and are as follows : — 



<|= |(o,i)+(o,2)+(o, 8 )+&c. | / -nn]p-[Ei]?'-[oTT]f"-&c.j ' 



dt = ~ { ("'O+C^+C^-f&c. } h +E3*+lE3tf+[EEA*4- &c; 

 '^= {(i,o)+(i, 8 )+(i, s )+&c.}?~ri3Z-ini]?'-[ld]f"-&c.; 



% =~ { ( 1 > >+( 1 ' 2 )+0' 8 )+ &c - } *+EII* +E3* , +E3*H- &c. ; 



&c. 



If we denote by e, e', e", &c, &, v>', ex", &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 -a, h'^e! sin ci', 7i"=e" sin cr", &c, ) .,. 



I ==e cos s, V =e' cos zj', 7" =e" cos ot", &c. j 



Whence we deduce 



e 2 =7t 2 -f-Z 2 , e'"=A' 2 -|-Z' 2 , e" 2 =7i" 3 -f-r 2 , &c ; tan c*==-j' tan E7'=,„ &c. (2) 



If 7i, I, h', V, &c, are determined by the integrals of equations (A), for any time 

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

 of e, e', &c, cr, gj', &c. 



May, 1871. \ ( 1 ) 



