( 759 ) 
this line is thrown to infinity, whilst y = nur passes into u = 1 : (m + 1). 
We find 
an uh 
du se u 
and finally from this 
,u—v=—lIgu+C 
To make use of the second pencil we determine a projective trans- 
formation, which transforms (4 + 1) : (w — 1) into a linear function 
of w and «=O into v=0. These conditions are satisfied by the 
substitution 
v u—v+1 y+l1 2u+ 
wa wale ar ute TT hin 
We now find 
: dv u du 
v Tul} 
3. When the linear equation 
y = Pe)jy + Qe) 
has a polar line we can transform this projectively into the line at 
infinity. The linear equation is then transformed into a homogeneous 
one, or, where this is not possible into an equation of the form 
ors 
on 
In the latter case w=:m is the critical pencil; in the former 
where 
dv an v 
du * \u 
the point «=O, v =O is the vertex of the critical pencil of rays. 
Let 
. at + by +e=0 
be the polar line of 
y= Pay + Qa), 
thus the locus of the pole 
If we put 
