THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 175 



If we put <=0, in equations (563) and (564), we shall get the values of pre- 

 cession and variation of obliquity at the beginning of 1850. These values are 



<Z 7 -=0".489682. 



tit ~ dt 



If we take the sum of the coefficients of the sines in the expression of 4' equation 

 (561), without regard to their signs, we shall get the maximum quantity by 

 which the true place of the equinox can diifer from its mean place. This sum is 

 14185".81=3 56' 25".81 ; it therefore follows that, if we suppose the equinox to 

 have a uniform yearly motion equal to 50".438239, its place, when computed for 

 any epoch, will not differ from the true place by an amount exceeding 3 56' 26". 

 This remark is of especial importance in regard to the computation of the elements 

 of terrestrial physics during past geological periods. 



If we divide the number of seconds in the circumference of the circle by the 

 mean motion of the equinoxes, we shall get 1296000"-^-50".438239=25694.8=the 

 number of years required for the equinoxes to perform a complete revolution in 

 the heavens. 



If we take the sum of the coefficients of the cosines in the expression of e 

 equation (562), without regard to their signs, we shall obtain the maximum 

 quantity by which the obliquity of the ecliptic can differ from its mean value. 

 This sum is 4720".96=1 IS 40".96. From this it follows that the obliquity of 

 the ecliptic is always confined within the limits 23 IT 16".571 18' 40".96; or, 

 between 24 35' 57".53 and 21 58' 35".61. The amount of its oscillations cannot 

 therefore exceed 2 37' 22". 



If we take the sum of the coefficients of -^, equation (563), we shall get the 



dt 



maximum quantity by which the annual precession can differ from its mean value. 

 This sum is equal to 2".225841 ; whence it follows that the annual precession is 

 always confined within the limits 



50".4382392".225841. 



The maximum value of precession in a Julian year is therefore equal to 

 52".664080, and the minimum value of precession during the same time is equal 

 to 48".212398. If we divide the difference of these two numbers by the time 

 required for the earth to describe one second of an arc, we shall get the maximum 

 variation of the tropical year, equal to 



4".451682-=-(3548".1876)-^86400 9 )=108 s .40 seconds of time. 



If we subtract the present value of the precession from its maximum value, we 

 shall get 2".42836 for the difference between them. Dividing this number by 

 the time required for the earth to describe one second of arc, we shall get the 

 amount of time by which the present tropical year exceeds the tropical year when 

 it is reduced to its minimum length. The number thus found is 59M3. In like 

 manner we shall find that the tropical year may exceed its present length by 



