7 
VIb. for the ellipse (A,, > 0) a,,4 < 0, thus 5 purely imaginary and 
en (Vv) 
n* (v) 
Bien the preceding we see that » must move in its complex 
plane on the sides of the rectangles of the net formed by the lines 
y= mk - purely imaginary and » = nik’ + real. 
g _ Aret =F Ars 
Das a 
also = 
purely imaginary. 
The value of 6? = +1 is evidently positive on 
33 
that side of the polar line =O of O with respect to the conic 
where 0 lies itself; on the other side $* is negative. The polar line 
g=O0 of O divides therefore the plane into two parts: in one (in 
which O lies) & is real, in the other ¢ is imaginary. 
In the points of contact A, and fF, of the tangents out of O to 
the conic § is 0, so /=o. 
In the points at infinity S, and S, we find that & and ¢ are both 
infinite and / is also equal to o. 
The diameter passing through O (A,,v—-A,,y—=0) intersects the 
conic in two points 7, and 7, for which 5— 0, thus J=0. 
If we substitute the expressions (84) for & and & in the formulae 
(78) we at last arrive at x and y as functions of t. 
With a view to VA,, being real or not, we shall deal with the cases 
of IV and VI separately. Farthermore we shall express 4 everywhere 
1—A 
in Salam thus in the anharmonic ratio of the four points 
RR, S,,S, We shall give the formulae for « only. The expres- 
sions for y we can easily find by replacing a,, in those for 2 by 
—a,, and A,, by A,, 
We then find at last: 
1—d 1-+en (pr) Aas A,; 
BENE 20 a ©) | - (1 4d) Per dn(v) + — — A dont | 
Ay; 
Veer 
BE ——-;3 
14-d 
_ 1 en(v)+dn(r) | ae Ads 
{Va 1 20 EG) ee | (1 +d) (ime 5 4- S97 {den(v) 4- ings) | 
1 
_ se 
1+d 
_ 1 en(v)+dn(v) le le cn | 
heer 25° isn (vp) ee Std AA AL ee eos | 
