( 778 ) 
three right lines, namely the three tangents which we can draw 
out of Y to the Cayleyana. 
This can be confirmed as follows. Let Z be a point of the locus 
of X, which is determined by 4) and X a second point of that 
locus lying on YZ, so that we have a x = lay + \ta z . Out of (4) 
then follows 
(abc) ajy (la y + (M z ) (lb y + yb 2 f = 0 . 
By exchanging a and c we see at once that 
(abc) a y c) ( 26 , + y.b z f 
vanishes identically. Analogously we find that {abc) a^ya z b) and 
(abc) ayCya z byb z vanish identically. As finally the form (abc) ayC y a 2 b z 
is zero because Z lies on the locus indicated by (4) the above 
relation is satisfied by all points of YZ, so the locus consists of 
three lines through Y. 
4, That the line g== XV is tangent to the Cayleyana as soon as 
(4) is satisfied, can be confirmed by reducing (4) to the tangential 
equation of that curve. In the first place we find out of 
(abc) axayb^Cy = 0 and (acb) ayflyc x by == 0 
the relation 
(abc) OjOy (bjfiy + byCx) (b x c y — byC x ) = 0. 
The last factor can be replaced by (beg) where & indicate the 
coordinates of XY. After that the equation can be broken up into 
two terms, which pass into each other when b and c are exchanged. 
So we can replace it by 
(a6c)a^/a%(^l)=0 - . r -.( 5 ) 
farthermore it is evident from 
(abc) (hflyblcy =r 0 and (c&a) c*c,6*ay == 0, 
that at the same time is satisfied 
(abc) b&yCy (ocg) = 0, 
(6ac) aJbyCy (beg) = 0.(6) 
By combining (5) and (6) we find 
(abc) o*c, (beg) (abg) = 0. 
So 
(abc) exay (beg) (abg) —: 0. 
Out of the last two relations follows finally 
