dt : 
If from this we solve —~, multiply by d%, and integrate from 0 
{ ld 
to 2a, we find the period of revolution 7. The result naturally 
depends on the point of starting, i.e. on the point on the ellipse 
where 9 == 0. In the moving ellipse the true anomaly is 
v= gd —w. 
If we take this as independent variable, we must integrate from 
v, to v, =v, + 2ay.- 1, find in this way 
na VAT OR GA: : 
eee — — 4+-—ecosl, +...], 
kym | 2a Pp Pp 
where /, is the mean anomaly corresponding to the true anomaly 
v, =4(v,+v,). All neglected terms of the series, as well as the 
last term which has been included, are periodic. If these are 
omitted, we find the mean period 7. If then we put 27, = 2a, 
ryt 
we find 
3h ‘ 
a’n® = k?m} 1 — — (1—3se’) Cite via a aaa 
P 
which replaces KePrer’s third law. 
Let the excentric and the mean anomaly in the moving ellipse, 
corresponding to the true anomaly v, be called w and / respectively. 
Then 
u —esinul 
EN Te. 4? (35) 
—_=n| Ì (1 -- 4e?) — | 
dt P ae 
The mean value of the expression within the square bracket over 
; 32° 
a complete revolution in the (moving) ellipse is 1 — — =g. 
P 
4. The motion of the moon. 
The moon will be considered as a material point, of which we 
will investigate the motion in the gravitational field of the sun and 
the earth. 
We take an arbitrary system of rectangular coordinates, in which 
x; , 4 are the heliocentric coordinates of the moon, 
Si bj Q 9 99 29 2 99 He) earth, 
Hir ioe VeoCentnic 4 nt ON 
Thus’ == — Ei cand: we put 
A Arn (7) ; vg ao AY (Sh 
m being the mass of the sun, and m, of the earth. We can neglect 
