( 1252 ) 
must be equivalent with 
(v? + 2H ay + By’ + 2Ga + 2F y+ C) (er + By) = 0. 
This may be done by chosing a = — (b° +c®), 2=0, 
c c: c c 
fe B=—, WE FS SS 
b b? b? c? (b? + c°) 
Hence a first particular integral is the conic 
c? 
bi y)? + 2cy + ——_ = 0 
(be + cy)* + YT Rie 
In the same way we find that the differential equation is satisfied 
by the curve of the third degree 
c? 
== 
+e 
Combining these, the general integral is found to be 
(ba Hey)? + dey (bz Hey) + dey + 
(be cy)? + Bey (beter) + Bey + 
c? 3 
=) —— Const. 
c 
e+ 
c? 3 
b? +-¢? 
which for small values of « and y may be expanded in the form 
e+tytFRr,+F,+...= Cons. 
Therefore the origin is also a centrum in this case. 
Resuming we may conclude that when the differential equation 
is reduced to the form 
x 
(ba-ey)* + 2ey + 
dy = —a+a'x’+ 2b'ay—a'y’ 
da yara? + 2bey ey? 
the origin is a centrum only in the four following cases 
15 a =6, and ‘atb == 
2. es (arte 
ae a’ =b, at+c=0 and the roots of 
aA*—3bA? + (2b'—a) A+6 = 0 all real. 
A, a’ = b, 2b'= 3a+5c and ac+6?+2c’ — 0. 
In all other cases the origin will be a focus for the real integral 
curves. 
