38 SECULAR VARIATIONS OF THE ELEMENTS OF 



For the planet Venus we have, 



Maximum e'=^'+i^'+i^ 2 '+&c. =0.0706329. One-half of this is 0.0353165. 

 As this number exceeds any one of the coefficients, N', 2%, iV 2 ', &c, it follows that 

 the perihelion of the orbit of Venus has no mean motion, and that the minimum 

 value of its eccentricity is zero. 



For the Earth we have, 



Maximum e"=i\T"+JV 1 "+iV 2 "+&c.=0.0677352. One-half of this is 0.0338676. 

 As this number exceeds any one of the coefficients N", N{, iV 2 ", &c, it follows that 

 the perihelion of the Earth's orbit has no mean motion, and that the minimum value 

 of the eccentricity is zero. 



For the planet Mars we have, 



Maximum e'"=iy'"+iVi'"+iV 2 '"+&c.=0.1396547. One-half of this is 0.0698274. 

 As this number is less than iV 3 '", it follows that the perihelion of the orbit of Mars 

 has a mean annual motion equal to g 3 or 17".7844562; and that the minimum 

 eccentricity of his orbit is equal to 0.0184753. We shall here observe that a small 

 variation in the assumed mass of the Earth would produce a considerable variation 

 in the limits of eccentricity and mean motion of the perihelion. 



For the planet Jupiter we have, 



Maximum e JF =i\^+iV 1 JF +iV 2 JF -|-&c. =0.0608274. One-half of this is 0.0304137. 

 As this number is less than N/ T , it follows that the perihelion of the orbit of Jupi- 

 ter has a mean annual motion equal to g 6 or 3".7166075; and that the minimum 

 value of the eccentricity is equal to 0.0254928. 



For the planet Saturn we have, 



Maximum e r =i\r r +iV 1 r +iV 2 r +&c 1 =0.0843289. One-half of this is equal to 

 0.0421644. As this number is less than N 7 V , it follows that the perihelion of Saturn's 

 orbit has a mean annual motion equal to g % or 22". 4608479 ; and that the minimum 

 value of the eccentricity is equal to 0.0123719. 



For the planet Uranus we have, 



Maximum e rJ =i^"+i^ rj +i^ F/ +&c. =0.0779652. One-half of this is 0.0389826. 

 As this number is less than JV 6 ", it follows that the perihelion of the orbit of Uranus 

 has a mean annual motion equal to g 6 or 3".7166075 ; and that the minimum value 

 of the eccentricity is equal to 0,0117576. 



For the planet Neptune we have, 



Maximum e F// =i^ FJ/ + iV 1 FJJ +i^ 2 r// +«&c. =0.0145066. One-half of this is 

 0.0072533. As this number is less than NJ n , it follows that the perihelion of 

 Neptune's orbit has a mean annual motion equal to g± or 0".6166849 ; and that the 

 minimum value of the eccentricity is equal to 0.0055712. 



21. We see, by the preceding article, that the mean motions of the perihelia of 

 Jupiter and Uranus are exactly equal, It follows from this circumstance that the 



