( 1242 ) 
(h? 4 k)A=la eht (a Hek 
0 == (a He) h — (a He) k 
SO 
ate ate 
ih a= AS Ae 
2 2 (1) 
2 being a real number whatever, except zero. 
From this it is evident that we may write 
dy ear +bey ay et 
de wy EE ax? EE 2bay + cy?’ ce ytx 
where still ¢ could be replaced by a — 2. As we do not want this 
condition we will retain this coefficient in the old form. 
Now after Porncark’s') theory here the origin is a centrum when 
it is possible to construct an infinity of homogeneous functions of 
order 7, satisfying the following series of partial differential equations 
N= 
aa ie aX + WY 
Lv oy nn U DE == UL y 
OF oF OF, oF 
ae eee ey (2) 
Oy Ou Ow Oy 
Mee Ee 
Cc === — = A — Ae 
Pia vr oe ae Fo 
This leads to an infinity of conditions for the five constants a, 4, c, 
a,b and if these are all fulfilled the origin is a centrum and the 
general integral may be written 
ety 4+Fi,4+F, + F,+...= Const. 
where the series converges until the closed curves, represented by 
this equation, pass through the nearest singular point. 
3. The equations (2) may be transformed as follows. If we suppose 
; OF, OF, 
F, to be a homogeneous function of degree n, and «———y 
es Ow 
eae es = = 
to be divisible by zX + y/Y, the function X aes oe will also 
dy 
be divisible by «X + 1X. For eliminating ai “difforential quotients 
between 
OF , 5 On, ° 7 4 
vu ~—-—y == (aX a yk ) Pr—s 
Oy Ow 
1) Journ. de Math. (1885) p. 173. 
