xii INTRODUCTION. 



less than 4° 41' ; while its inclination to the invariable plane of the planetary- 

 system always oscillates within the limits 0° 0' and 3° 6'. It is also evident that 

 the mean motion of the node of the apparent ecliptic on the fixed ecliptic of 1850, 

 and also on the invariable plane, is wholly indeterminate. 



The mean value of the precession of the equinoxes on the fixed ecliptic, and 

 also on the apparent ecliptic, in a Julian year, is equal to 50".438239 ; whence it 

 follows, that the equinoxes perform a complete revolution in the heavens in the 

 average interval of 25,694.8 years; but on account of the secular inequalities in 

 their motion, the time of revolution is not always the same, but may differ from 

 the mean time of revolution by 281.2 years. We also find that if the place of 

 the equinox be computed for any time whatever, by using the mean value of pre- 

 cession, its place when thus determined can never differ from its true place to a 

 greater extent than 3° 56' 26". The maximum and minimum values of precession 

 in a Julian year are 52".664080 and 48".212398, respectively, and since the length 

 of the tropical year depends on the annual precession, it follows that the maximum 

 variation of the tropical year is equal to the mean time required for the earth to 

 describe an arc which is equal to the maximum variation of precession. Now this 

 latter quantity being 4".451682, and the sidereal motion of the earth in a second 

 of time being 0".041067, it follows that the maximum variation of the tropical 

 year is equal to 108.40 seconds of time. In like manner, if we take the difference 

 between the present value of precession and the maximum and minimum values 

 of the same quantity, we shall find that the tropical year may be shorter than at 

 present by 59.13 seconds, and longer than at present by 49.27 seconds. We also 

 find that the tropical year is now shorter than in the time of Hipparchus, by 

 11.30 seconds, 



The obliquity of the equator to the apparent ecliptic, and also to the fixed 

 ecliptic of 1850, has also been determined. The variations of this element follow 

 a law similar to that which governs the variation of precession, although the 

 maximum values of the inequalities are considerably smaller than those which 

 affect this latter quantity. The mean value of the obliquity of both the apparent 

 and fixed ecliptics to the equator is 23° 17' 17". The limits of the obliquity of the 

 apparent ecliptic to the equator are 24° 35' 58" and 21° 58' 36"; whence it follows 

 that the greatest and least declinations of the sun at the solstices can never differ 

 from each other to any greater extent than 2° 37' 22". And here we may mention 

 a few, among the many happy, consequences which result from the spheroidal form 

 of the earth. Were the earth a perfect sphere there would be no precession or 

 change of obliquity arising from the attraction of the sun and moon ; the equinoc- 

 tial circle would form an invariable plane in the heavens, about which the solar 

 orbit would revolve with an inclination varying to the extent of twelve degrees, 

 and a motion equal to the planetary precession of the equinoctial points. The 

 sun, when at the solstices, would, at some periods of time, attain the declination 

 of 29° 17' for many thousands of years; and again, at other periods, only to 17° 

 17'. The seasons would be subject to vicissitudes depending on the distance of 

 the tropics from the equator, and 'the distribution of solar light and heat on the 

 surface of the earth would be so modified as essentially to change the character of 



