THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
821 
The period of the precession of the two proper planes is —27r//c 2 , and that of the 
precession of the two nodes on their proper planes is 2 tt/(k. : — k 1 ). 
In the preceding computations we omitted a common factor rt/n from a, /3, a, b, 
k v k . 2 ; this factor must now be reintroduced, r is a constant and log r'= 177242, 
then by means of the numerical values given in the first table I find 
£ = 1- -88 76 
log r£/n+10 = 7-80940 879708 8-62750 
Also 
lo g—^+10 = 9-99401 9 89462 973295 
log (/Co — k { ) is given before in Table II. Then introducing the omitted factor rt/n, 
I find 
g = 1- -88 -76 
— 27r//c 3 =988 yrs. 509 yrs. 434 yrs. 
2tt/(k. 2 —/Cj)= 60 yrs. 77 yrs. 51 yrs. 
Thus both precessional movements on the whole increase in rapidity (because of the 
increasing value of rt/n), but the rate of the precession of the pair of proper planes 
increases all through, whilst that of the precession on the proper planes diminishes 
and then increases. It was pointed out towards the end of § 13 that /c 2 is, so to speak, 
the ancestor of the luni-solar precession, and k 2 — k { the ancestor of the revolution of 
the moon’s nodes. Hence the 988 years has bred (to continue the metaphor) the 
present 26,000 years of the precessional period, and the 60 years has bred the present 
18-g- years of the revolution of the moon’s nodes. 
We see that the k 2 — k 1 precession attains a minimum at a certain period being more 
rapid, both earlier and later. 
All the above results will be collected and arranged in a tabular form, after further 
results have been obtained by means of an integration, carrying out the investigation 
into the more remote past. 
The tidal and precessional effects of the sun’s influence have now become exceedingly 
small, and the only way in which the sun continues to exert a sensible effect is in its 
tendency to make the nodes of the lunar orbit revolve on the ecliptic. In the analysis 
therefore we may now treat t as zero everywhere, except where it occurs in the form 
r'/Xtr. Since X and C are both pretty small, these terms in t'/t rise in importance. 
The equation of conservation of moment of momentum now becomes 
-=1+7(1—f) 
Here kn 0 is equal to the value of lit in the preceding integration when 7= 76 ; and 
hence l/kn 0 = '665903. 
Then we now have /3=b, y=g, S=d, /3'=b', y'=g', S'=cT, but a and a' are not 
equal to a and a'. 
MDCCCLXXX. 5 N 
