( 1241 ) 
Mathematics. — “On the centra of the integral curves which satisfy 
differential equations of the first order and the Jirst degree.” 
By Prof. W. Kaprryn. 
1. Considering and y as the coordinates of a point in the plane, 
the real curves which satisfy a differential equation of the form 
dy = @ 
dx P’ 
present different singularities. Between these we meet with points 
(foci, which are asymptotic points for the integral curves which 
present themselves as spirals in the neigbourhood of such points. 
Q and P being polynomia in # and y with real coefficients, 
These spirals sometimes change in closed curves and then the corre- 
sponding focus is called a centrum, and it is a question of great 
interest to determine the conditions when this happens. This question 
has been solved theoretically by Potncaré, but meets with great 
difficulties in practice. 
The object of this paper now is to examine the differential equa- 
tion, supposing P and Q to be polynomia of the second degree, and 
to determine all cases when centra may be expected instead of foci. 
2. When the origin of coordinates is the point which must be 
examined, the differential equation may be written 
dy —e haa? + Way Hey? 
de yv oe aa? + 2bey +- cy? ; 
where a,b. c, a',b', c', are real constants. 
By substituting 
§ = he + ky n= — ke + hy 
the form of this equation is not changed, for we get 
| dy —§ + ast + En + yi 
dg H+ ast 2B5n + yr? 
where 
(2? + kh? a =ah? + (a + 26) hk + (2B' 4+ c) Ak? + eh? 
(AP + kP B = bh? — (a -— b' — ohh — (al +b —c)hk? — O'R 
(A? + k*)? y = ch® — (2b—c’) hk + (a — 20') hk? 4+ ak? 
(2? + kK’)? a = ah? — (a — 20') hk — (26 —c’) hk? — ch? 
(A? + k)? B= Wh? — (a Hb — el) hk +a — B' — e) hk? — bk? 
h? +k’)? y' = ch? — (2b' He) h°k + (a! + 2b) Ak? — ak? 
Now / and & may be chosen so that the six coefficients ce @ y, 
«By satisfy two conditions. Adopting 
aty=a aty=—0 
we have wide 
