( 757 ) 
A pencil of rays with the indicated property I call critical. 
From the following ensues the property : 
When a singular point of a differential equation of the first order is 
the verter of a critical pencil of rays, then this equation can be 
reduced by a projective transformation into a linear one. 
tata ty — 7, is the vertex of, a ,eritical. pencil, then thé 
u 
substitution 
ji 
a U Ui en 
Vv Vv 
leads to the aim in view. 
ExampLn I. 
The equation 
wv + 2e?y + ay*® + y* 
dy iy 
ae + ary? + sy? 
da 
has in e=0, y=0 a singular point. 
The tangent in a point of the line y= me is indicated by 
(m +. 1) ms + 2m + 1 eg 
(m+ 1) mx + 1 
Y — me = 
3y reduction it is evident that the parameter 2 appears here only 
linear; so the pencil is critical. 
For the locus of the poles we find the cubic curve 
2m + 1 
1 
; ee (m + 1) mm 
z= — varen ra 
(m+ 1)m? * 
By the substitution 
1 u 
bs ie 
Vv Vv 
y= me passes into w= m, and the given equation into 
isu “+ = deu 
du u+ 1 
2. Let us treat in particular the equation 
dy 
rae P(x)y . 
Here the locus of the poles of z = 1m is the line y = 0 (polar line). 
As the homogeneous equation 
polar line, the question arises whether it is possible to transform the 
