( 779 ) 
(abe) (acQ (beg) (abg) — 0.p) 
This tangential equation really represents the Cayleyana »). 
So we have found that the six polar lines p x ,p y ,p z ,p xy ,p r ,p zx 
concur in one point when the points X, Y, X lie on a tanqent of 
the Cayleyana. 
5. When p x , p y , p 2 are concurrent we have 
(abc) atblct^zO .(8) 
This equation gives thus the relation between the coordinates of 
three points lying on one and the same polar conic. 
For an arbitrary choice of X and Y this equation is satisfied 
except by X and Y by no point of the line XY. If it is to be 
satisfied by zk = X.vk -j- \tyk we must have 
(abc) £ £ (Xc x + p c y y = 0, 
therefore 
XfJt (abc) a~ x by c x c y — 0. 
This is satisfied for each value of X.p whfen the relation (4) is 
satisfied, so when X, Y, Z lie on a tangent of the Cayleyana. 
Now in general the polar \a\esp xy ,p xz , p yz form a triangle inscribed 
in the triangle pxPyPz (see § 3). If (4) is satisfied then p xy , p X2 ,p yz 
are concurrent; but then their point of intersection must be at the 
same time point of intersection of p x , p y , p z . 
If X, Y, Z are three collinear points of the cubic, then p yz , p zx 
and Pxy pass successively through X, Y, and Z. 
For, from a\ = 0, a] — 0 and (Xa x -f- pa y f = 0 follows that the 
point Z is indicated by a&y (Xa x -j- pa y ) = 0. So we have a x a 9 a z = 0, 
so Z lies on the polar line p xy . 
If moreover X, Y,Z lie on a tangent of the Cayleyana , then 
Py*> Pzx, Pxy must coincide with the tangents p x , p y , p z in X,Y,Z. 
6. For p x , p y , and p„ y to be concurrent, there must be a point W 
for which we have a£z w = 0, £h w = 0, and c x CyC w = Q. 
But then (abc) aj£/c x c y = 0. 
For arbitrarily chosen Y the locus of X becomes a figure of the 
third order, passing through Y, because we have (abc) ap££0Q. 
But by taking notice of (4) we see that this figure consists of three 
tangents of the Cayleyana. Out of 
a£z* = 0, a y a m = 0 and OjOyOw = 0 
») See e.g. Clebsch, Lemons sur la geometrie, 11, p. 284. 
