( 756 ) 
Mathematics. — “A family of differential equations of the first 
order.” By Prof. JAN DE VRIES. 
(Communicated in the meeting of February 27, 1909). 
1. The tangent in (x,y) to the integral curve of 
dy 
ae P(@)y + QW) 
is represented by 
Y —y={P(«)y + Q(#)} (X — 2). 
For the points of the line «=m we have thus 
{ Qn(X — m) — Y} 4 y{ Pan(X — m) + 1} =0, 
The tangents indicated by them form therefore a pencil of rays 
having the point 
1 Qn 
Cie P, 
as vertex. 
I call this point the pole of the line «=m. 
By a projective transformation the linear equation is transformed 
into a differential equation, determining a pencil of rays of which 
each ray has a definite pole. Each line, connecting the vertex 
S of that peneil with a pole, having to touch in San integral curve 
NS is a singular point, i.e. a point where y’ is indefinite. To confirm 
this we transform the linear equation by the substitution 
au+av-a, & butbwvt ob, B 
oa ee cu Ae ee 
On account of that 
y' =P (x) y + Qa) 
passes into | 
dv (ay — c,0) (BP* + 7Q*) — (by — «By 
du (by — 68) — (ay — e,a)(BP* + 7Q*) 
p=), Qe — o(<). 
7 y 
The pencil «=m is then transformed into the pencil of rays 
where 
a=my- 
F Ei er men 
or a=0, y= 0 we really fin mn 
hj Also the points 6 =0, 7 =0 and y=9, P*=0 are singular. 
