213 
roots in a series of ascending powers of a, the terms of degree 0 
and 1 do not change any longer. From (36; it follows in this way 
4w, = Za (l + 3 am) =2r + Jama. 
Now (85) shows that m —n is the smallest, m + n the greatest 
l : 
value of —. From this or from (85a) it follows that m is the 
qr 
reciprocal value of the parameter p of the orbit and n/m represents 
the excentricity ; so 
— 4 PP == e ° e e e 5 e . (37) 
In m 
This gives for the motion of the perihelion per period 3ar/p cor- 
responding to the value calculated by Ernsruin. 
To conelnde we will calculate the periodic time. From (14) follows 
r* dp 
me oe 
a 
ien 
r 
If we put in this «a =0 we obtain the corresponding equation of 
Nrwton’s theory; we may therefore expand the denominator and 
obtain as a first approximation 
Yo amy (: + =| apr dg —--.ordg.. - i). (a8) 
r 
We must now substitute for r the value taken from (35). Let us 
for a moment introduce the elliptic function sn with the modulus 
k, defined by 
a a Zan : (39) 
e,—e, 1—d8am-+an 
(35) passes into 
] oe 
ae am On sn ko Wie, —e, ee (40) 
A? is of the first order, and consequently very small. If we put 
mA sn speen . 6.0. « N41) 
we find by differentiation 
cos dy = 4 Ve,—e, V(1 — sin? w) (1 — kh? sin? w) dy 
or 
dw 
a 
p ray VA? sin? wp 
Now as (40) passes into 
— = m — n + Zn sin’ Ws, 
Vd 
(38) becomes 
