THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
813 
Combining these four values by the rules of the calculus of finite differences, we 
have 
= •06641 
This is equal to log e sin^"—log e sin/ 0 . Taking j 0 = 5° 9', I find j= 5° 30'. 
Second period oj integration. 
From ^=1 to 76, four equidistant values were computed. 
From the computation for § 17 “Precession,” I extract the following :— 
£ = 1 -92 -84 -76 
, n 
log-se^ + 10 = 8-56746 879743 8-65002 872318 
° n 
Then assuming 5° 55' as an average value for j, I find 
f = 1 -92 -84 76 
2^1— ~ secisec/^j ] = -5193 *5660 '6232 *6948 
Combining these, we have 
This is equal to log,, sin j —log* smj 0 . Taking j 0 = 5° 30 / from the first period, we 
find j = 6° 21'. 
This completes the integration, as far as it is safe to employ the methods of Part II. 
In Part III. it was proved that, in the case where the nodes revolve uniformly, 
equations (224) reduce to those of Part II. But it was also shown that what the 
equations of Part II. really give is the change of the inclination of the lunar orbit to 
the lunar proper plane ; also that the equations of “ Precession ” really give the change 
of the inclination of the mean equator (that is of the earth’s proper plane) to the 
ecliptic. 
MDCCCLXXX. 5 M 
