follows indeed 
(fax -J- ^ ’ 
i.e. if ^ lies on IF, then p z will pass through the point of inter¬ 
section W of p x ,p y and p xy , which bears then at the same time 
p yz and p xz > 
So the three lines p x , p y , and p xy concur only then in one point 
when X and ¥ are united by a tangent of the Cayley ana. Their 
point of intersection bears then also all the polar lines and mixed 
polar lines belonging to the points of those lines. 
The lines p a , p ty , and p xz will be concurring, when 
(abc) a^bJtyCxCz — 0 
is satisfied, thus also 
(a&c) alwybzh = 0, 
hence also 
If we put 
(aic) cficbxCj; ( b y c z — b z c y ) = 0. 
ykzi — yi zjc = £mt 
we have the condition 
(«6e)a'v,(^ = °- 
As this can also be written in the forms 
(abc) aJhfix (ac£) = 0 and (abc) a x b x c x (a&£) = 0 
and as out of 
b l b 2 b t b x 
g, g, I, & 
follows the relation 
(abc) £*=:«* (bc§) + b x (co£) + c x (ab%) 
the above condition can be replaced by 
(a^) ! «M = 0. 
With arbitrary position of X this is satisfied by £* = 0, i.e. when 
X, Y, and Z are collinear (see § 3). 
If however 
(a&c) 2 aJ>xC x = 0, 
so that X lies on the Hessian, then X, Y, and iT are quite arbitrary. 
This was to be foreseen, now namely n x is a pair of lines, so that 
the lines p x , p xy , and p xz concur in the node of n x . 
