SECULAR VARIATIONS OF THE PLANETARY ORBITS. 2G5 



eoceutricities and incliuatious relate cliiefly to tlieir inngiiitiidb at any 

 time; but we ma}" also desire to kuow their rates of cliaiige at any time, 

 aud the limits within which they will perpetually oscillate. In regard 

 to the nodes and perihelia, it is sometimes necessary to know their rela- 

 tive positions when referred to any plane and origin of coordinates ; 

 aud also their mean motions, together with the amount by which tlieir 

 actual places can differ froai their mean places. Witli respect to the 

 magnitudes and iiositious of the elements, together with their rates of 

 change, we may observe that our equations will give them for any 

 required epoch, by merely substituting in the formulas the interval of 

 time between the epoch required and that of the formulas, which is the 

 beginning of the year 1850. An extended tabulation of the variations 

 of the elements does not come within the scope of our work ; and we 

 leave the comi)utation of the elements for special epochs to those inves- 

 tigators who may need them in their researches. We shall here give 

 tlie limits between which the eccentricities and inclinations will alvrays 

 oscillate, together with the mean motions of the perihelia and nodes 

 on the fixed ecliptic of 1850 ; and shall also give the in(;linatious aud 

 nodes referred to the invariable plane of the planetary system. 



For the planet Mercury, we find that the eccentricity is always included 

 within the limits 0.12111)43 and 0.2317185. The mean motion of its 

 perihelion is 5".4G3803 ; and it i^erforms a complete revolution in the 

 heavens in 237,197 years. The maximum inclination of his orbit to the 

 fixed ecliptic of 1850 is 10° 30' 20", and its minimum inclination is 

 3^ 17' 8" ; while with respect to the invariable plane of the planetary 

 system, the limits of the inclination are 9^ 10' 41" and 4P 44' 27". The 

 mean motion of the node of Mercury's orbit on the ecliptic of 1850, and 

 on the invariable plane, is in both cases the same, aud equal to o".12G172, 

 making a complete revolution in the interval of 252,823 years. Tiie 

 amount by which the true place of the node can differ from its mean 

 place on the ecliptic of 1850 is equal to 33^ 8', while on the invariable 

 plane this limit is only 18^ 31'. 



For the planet Venus, we find that the eccentricity always oscillates 

 oetween and 0.070G329. Since the theoretical eccentricity of the orbit 

 of Venus is a vanishing element, it follows that the perihelion of her 

 orbit can have no mean motion, but may have any rate of motion, at 

 different times, between nothing aud infinity, both direct and retrograde. 

 The position of her perihelion cannot therefore be determined within 

 given limits at any very remote epoch by the assumption of any par- 

 ticular value for the mean motion, but it must be determined by direct 

 computation from the finite formulas. The maximum inclination of her 

 orbit to the ecliptic of 1850 is 4° 51', and to the invariable plane it is 

 3° 1G'.3 ; while the mean motion of her node on both planes is indeter- 

 minate, because the inferior limit of the inclination is in each case 

 equal to nothing. 



A knowledge of the elements of the earth's orbit is especially inter-" 

 esting and important on account of the recent attempts to establish a 



