( 1245 ) 
The origin is therefore a centrum if both the conditions (5) and (6) 
are satisfied. 
In this case the integral of the differential equation 
dy = — #+b2? — ary — by” 
des y+. aw? Qbay+-cy? 
takes the finite form 
vw + yy? + F, = const. 
where #, may be determined from (3). 
The integral curve 
oo a ee a be + 2aa7y + 2bay? +- er == const. 
thus represents a series of closed curves round the origin of coordinates. 
5. Assuming now 
a’ = b and a+ b’==0 
we may omit the factor a + 6’ and write 
P, = bay + (atb) y? = 9, ay + 9, 9". 
Now it is always possible to find a homogeneous function 
Po =7, a + 7, ay + 1, ey? dry 
satisfying the condition 
me gs ec ya & a)? 
Oy Ox Ow Oy Oa dy) * 
and the coefficients are found to be 
b 
rs eae Meares) 
in 
r, = b(2a + 4b' — c) 
1 
ry = 3 (2a* + LOal + 30° + 126). 
Proceeding to . 
P,=s, rt Heey +3, 2° y? + 8, vy? + 8, y* 
we find that the following relations between the coefficients of P, 
and P, must exist 
ie — (5a-+2b') r, + br, 
2s, — 4s, — 6br, + 4 (a+b) r, + 2dr, 
38, — 3s, = der, + Jhr, + (Bad-6b') r, + 3867, 
As, --28, = 2er, + (2a+80b') 7, 
-— s,= er, — 3br, 
