( 760 ) 
then 
afm + ¢ 
blm) 
and the linear equation obtains the form 
ble —f(@))y = by + af(«) + e. 
It is clear, that by the substitution 
ee ‚LZ 
u u 
Oi Tr 
the polar line is brought to infinity and the point of intersection 
of the rays «=m is brought into the origin. 
After some reduction we find indeed the homogeneous equation 
dv ae 
This reduction is apparently of no use when the polar line has as 
equation «+c=0O. Then 
1 
SS SS 
FP, m 
and the linear equation has this form: 
=S + Q 
mene" (w). 
By the substitution 
I v 
fe. en Eden 
u U 
the polar line is transformed into the line at infinity, the pencil 
“=m into the pencil wu=1:(m-+o), and we find the divided 
equation 
L du : 
dv+Q{——c}]—=0. 
u u 
4+. Let us look at a few more examples. 
ExamPLeE Ill. The equation 
" dy an yy +2 
da w?—xwy--2 
has singular points mest ys Ad; ¢ ='—1, y=1 and inde 
point at infinity on & = y. 
The pencil «—y=™m is critical. We find for the tangent 
(my + m*—2)(¥ —y) = (my + 2)(X—y—m), 
and out of this the polar line «+ y= 0. 
