INTRODUCTION. xi 



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

 the limits 0.1214943 and 0.2317185. The mean motion of its perihelion is 

 5". 463803; and it performs a complete revolution in the heavens in 237,197 years. 

 The maximum inclination of his orbit to the fixed ecliptic of 1850 is 10° 36' 20", 

 and its minimum inclination is 3° 47' 8" ; while with respect to the invariable plane 

 of the planetary system, the limits of the inclination are 9° 10' 41" and 4° 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, and equal to 5". 126 17 2, making a 

 complete revolution in the interval of 252,823 years. The amount by which the 

 true place of the node can differ from its mean place on the ecliptic of 1850 is 

 equal to 30° 8', while oh the invariable plane this limit is only 18° 31'. 



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

 and 0.0706329. 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 and 

 infinity, both direct and retrograde. The position of her perihelion cannot there- 

 fore be determined within given limits at any very remote epoch by the assumption 

 of any particular 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° 16'. 3 ; while the 

 mean motion of her node on both planes is indeterminate, 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 interesting and 

 important on account of the recent attempts to establish a connection between 

 geological phenomena and terrestrial temperatures, in so far as the latter is modified 

 by the variable eccentricity of her orbit. The amount of light and heat received 

 from the sun in the course of a year depends to an important extent on the eccen- 

 tricity of the earth's orbit ; but the distribution of the same over the surface of 

 the earth depends on the relative position of the perihelion of the orbit with 

 respect to the equinoxes, and on the obliquity of the ecliptic to the equator. These 

 elements are subject to great and irregular variations ; but their laws can now be 

 determined with as much precision as the exigencies of science may require. We 

 will now more carefully examine these elements, and the consequences to which 

 their variations give rise. 



As we have already computed the eccentricity of the earth's orbit at intervals 

 of 10,000 years during a period of 2,000,000 years, by employing the constants 

 which correspond to the assumed mass of the earth increased by its twentieth part, 

 we here give the elements corresponding to this increased mass. We therefore 

 find that the eccentricity of the earth's orbit will always be included within the 

 limits of and 0.0693888 ; and it consequently follows that the mean motion of 

 the perihelion is indeterminate, although the actual motion and position at any 

 time during a period of 2,000,000 years can be readily found by means of the 

 tabular value of that element. The eccentricity of the orbit at any time can also 

 be found by means of the same table. 



The inclination of the apparent ecliptic to the fixed ecliptic of 1850 is always 



