214 
dw ad 
LBV e,-e, dt= ans ee! 
‘ aa (m-n + 2nsin? w)? V 1-k?sin2yp (m-n + 2nsin? Wp) V1-k sin? yp 
If «—=k=O0 we pass into NEwroN’s theory. Soin the first fraction 
we may expand the denominator and neglect £*, etc., and in the 
second fraction we may put £—=0. Putting 4° = Zan in the first 
fraction we obtain 
1+ ansin? wp adt: 
iB Ve,—e, | ee —- dw + - 
(nm —n+ 2n sin? yw)? m n+2nsin? wp 
1] —ta(m—n) jadw 
hp + 
aah 5 . (42 
(m—n-+ 2n sin? yw)? é (42) 
mn 2x sin? wp 
From the values of z,, z,, v7, we get, considering (22), 
rena ie 1—ds3am-+an 4 
tt: 2m (, — 2am S| 
We may write 
2m 
B 1(@—4)-= | [1 + 4 a(m — n)| 
at 
and so (42) passes into 
a d Zad 
ee le 
2m (m—n+-2nsin? ww)? mn 2nsin® pp 
We will call the time in which 7 is periodic the periodic time; 
it is the time in which y increases by 4, and wp by a. So 
il bals Tr - dw 43 s dip 
$ eee ee = 
; 2m (m —n + 2nsin® wp,” m—n + 2n sin? wp 
0 
0 
am Zan 
Etn ae 2 
(mtr) (mn?) 
In connection with (37) we get from this, a representing half the 
major axis: 
PE ae i 
—_— [T—a? + 2aa?, 
Qn 2 ; 
or with the same degree of approximation 
es DS leo ej. 
nva 
We so obtain instead of KeePrLeERr’s third law 
(a + a)’ a 
LTE 
We can also ask after the time required by p to increase by 22. 
This time depends on the place from which the planet starts; it is 
(43) 
