kK 
50 
may be expanded for small values of # and y in the form 
tte Waker ee ot == ONST: 
which proves again that the origin is a centrum. 
11. Resuming we may conclude that where 
(ate? + (al Hc)? = 0. 
the differential equation 
dy __—e dar + bay + cy? 
de y+ aa* + Qbay + a 
has a centrum in the origin of coordinates only in the following cases 
fig Spend) 
IId + ¢ = 240+ 0 sen atb —0 
HIT, 2a = + i(a— 2b' He), 26 => Eila + 2 He), 26= Hi (a—o) 
IV. of = te, od = + 1a, 26'= 3a4- 5e, Uda + de 
for it is easily seen that in the last three cases everywhere 7 may 
be replaced by — 2. 
The results obtained in our former paper show that the origin is 
also a centrum in the three following cases 
Ve Ae 0h = One ai 
Kl sa 2 ee 0, ae = 
Vide ea) x0! = De Ee AE ae alle 
We found there one case more viz. 
ae == Opava. an dede == 
but this is included in I. 
12. To compare these results with those of Dunac, we will 
transform our differential equation 
dy —ata'e’?+2b'eytcy?  —a+Y 
de 7 y + aa’ + 2bay + cy? By cee x 
in his form. This may be done by the substitution 
h§ =a + wy kn = & — wy. 
This gives 
hdé En kdn 
VAA Hil?) y+X—i(—#+¥) 
where 
y — te = — th3,- vs eN 
X 4 7i¥ = —i(A—B) MB — 21(C—C’) hk&n -- 1(D—E) kt 
X ~-1¥= —i(D+ EB) WS? — 21 (C4 C) Aken — (AFB) hea?’ 
and 
