used by the Ancient Egyptians. 183 



the cycle. They do this to such an extent, that 300 years will be found to be 

 the true length of the smaller cycle, and not 301, as would be the case if we 

 used the mean tropical year. The first of these causes is the annual change 

 undergone by the equation of the centre, proper to the point in the orbit where 

 the sun is situated at the commencement of the year. The sun's perigee passed 

 through that point in the orbit about 400 years before the chronological epoch 

 of the eighteenth century before Christ ; whence it is easy to see that for a long 

 course of ages about that epoch the sun would at the end of a mean tropical year 

 be behind his place at the beginning of it ; as the annual change in the equation 

 of the centre would always lengthen the year.* The other cause of the year being 



* Let 9 be the sun's longitude at the commencement of any year, reckoned from the mean 

 equinox of that time, and not corrected for lunar or planetary perturbations. Let fl' be the sun's 

 longitude, reckoned in like manner, at the end of any time t. The elliptic theory of the planets 

 gives us the following equations, n expressing the mean motion in longitude during that time in 

 reference to the mean equinox ; 



9=:e + 2esin(e— «T)-f.&c. (1) 



ff=znt + e+2e'sm(nt+£ — m')-\-kc. (2) 



The remaining terms of these values, containing the second and higher powers of the eccentricity, 

 may be disregarded ; as it is evident they can only modify in a very shght degree the results 

 obtained from considering the two first terms. At the end of a tropical year 



9'— e = 2s-; (3) 



and the value of t which satisfies this equation is, of course, the length of the tropical year. What is 

 called the mean tropical year is the value of t, obtained by leaving out of consideration the part of 

 the orbit in which the sun was situated at the beginning of the year ; or, in other words, by consi- 

 dering only the^rst terms in the above values, which are independent of the angle e — ■a. In the 

 mean tropical year, 8' — 6=:nt; and therefore, by (3) 



nt=2it; or t = ~. (4) 



n ^ 



It is evident that this value of t would abo satisfy (3), taking into consideration the other terms 

 in the values of 9 and 9' ; provided only that e and w were invariable. The divergency, then, of the 

 various tropical years that may be observed from the mean tropical year is due to the secular 

 variations of these elements. We know that e is continually diminishing, while ■m is continually in- 

 creasing. Let e — Se and iii-\-Sit express the values e' and jr', belonging to the end of the year ; 

 and let St be the variation of the length of the tropical year, caused by the variations of the elements. 

 It will obviously be a function of Se, Svr, and of the angle g — -ir; and it will depend on the magni- 

 tude of this angle (that is, on the part of the orbit where the sun is situated at the commencement of 

 the year) whether it is to be added to the mean tropical year, or subtracted from it. 



Substituting in (2) their values for e' and tj-', and writing t-\- St for t; confining ourselves abo 



I 



