212 
bolic, if 2 =m with the quasi-parabolic. The constants m and 7 take 
the places of 2 and u. We obtain 
e‚, —%— 2am, 
é, = —t+a(m+n), kind ae NCN 
e= — } + e«(im— n) 
In (22) and (26) we now must take, in the case of elliptic motion, 
is=w,, as p increases indefinitely, z remaining finite. In the case 
of the parabolie and hyperbolic motion 7 becomes infinite and so 
z—— +4; < moves between ¢, and e, and again is =w,. So (26) 
becomes 
a 
eri =t PEP + ,). 
Now we have the formula 
P(4 y +oj=e, Ae (e,—¢,) (e,— és) 
Ps a 
and so 
a ; (e,—e,) (e,—é) 
ssl PES IN GP yi ie 
r P 3 Pes 
or from (34) 
= ye 35 
Sas n+ rn, (35) 
This is the equation of the orbit required. If we now let « become 
“zero, e‚ and e, coincide, e,—e, becomes 1, and the /-function dege- 
nerates. We then obtain 
= = mn + Ansin? tL p—=m—neosgp . . . (354) 
and this equation shows once more that, if @a==0O, forn < m the 
motion is (quasi-jelliptic, for m >> (quasi)-hyperbolic, for n= m 
(quasi)-parabolic. For n = 0 it is circular, also if a is not supposed 
to be zero. The elliptic case is case 12 C, the hyperbolic is 12 B, 
the parabolic is 12 B, e, being supposed to be -—% there. 
14. Let us now examine the motion of the planets a little more 
in detail. Equation (85) shows that 4m, is the period; as the P-func- - 
tion is almost degenerated we may take 
4a 
Ve, ete 
dt 
A further approximation is not necessary as, after expanding the 
