38 SECULAR VARIATIONS OF THE ELEMENTS OF 



For the planet Venus we have, 



Maximum e'=JV+JV 1 '+A r 2 '+&c.=0.0706329. One-half of this is 0.0353165. 

 As this number exceeds any one of the coefficients, N', NJ, NJ, &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"=JV"+ J tfi"+Ay'+&c. =0.0677352. One-half of this is 0.0338676. 

 As this number exceeds any one of the coefficients N", N^', N 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 t r=2r+N l "+W+&c. =0.1 396547. One-half of this is 0.0698274. 

 As this number is less than NJ", 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' r =JV /r +A r 1 /r +JV/ r +&c.=0.0608274. One-half of this is 0.0304137. 

 As this number is less than N 6 ' 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 =W r +Jv7-f-JV 2 ''-f &c,=0.0843289. One-half of this is equal to 

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

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

 value of the eccentricity is equal to 0.0123719. 



For the planet Uranus we have, 



Maximum e ri =N v/ -\-N 1 "-\-N 2 r '-\-&c. =0.0779652. One-half of this is 0.0389826. 

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

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

 of the eccentricity is equal to 0.0117576. 



For the planet Neptune we have, 



Maximum e r "=N r "-\-N l r "-}- N 2 v "-\-&c. =0.0145066. One-half of this is 

 0.0072533. As this number is less than JV/", it follows that the perihelion of 

 Neptune's orbit has a mean annual motion equal to g 4 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 



