1008 
- OL 0 
L — r — — Pp — = constant 
Or Op 
and the equation | 
OL 
— = constant 
og 
are first integrals, which together can replace the equations of 
motion. If we call the first constant / and the second Ah, then 
w 
oh (17) 
Vw + ur? + vr? p' 
and 
—v 
ep =A: 
w 
<4 (RR) 
By these two equations p and 7 are given as functions of ¢; 
(18) presents close resemblance to KerPrer’s second law. 
Eliminating y from (17) and (18), we find 
apy rn Ar 
ul — | — w* | — —_—- ]—w 
dt h? 
vr” ; 
by which r 
is defined as a function of ¢; (18) then gives p as a 
function of f. 
In the case that the orbit just extends into infinity, r dr p*, 
and also ur? + vr? g* must be zero for r=—= ce, hence h—c accord- 
ing to (17). If h<ec, then 7 remains finite, and if h >c, the 
velocity is different from zero also for infinitely increasing 7. 
The orbit may also be circular; as in virtue of (18) y is constant 
in this case, dL/or will be constant, and the first equation (16) 
shows that 
OL ‘ 
‘an 1 
RENE: 
dw = d 2 0 
— gp? — (vr?) = 
dr Sa dr (or) 
by which the angular velocity is determined as a function of 7. 
fe 
In order to examine closer the motion of a particle we 
make use of the approximations for wu, v, and w, found above. If 
we put in (17) 
== Ts 
2k 
w——c?{ 1 — —}], 
er 
we get, expanding the root, 
