( 488 ) 
This result is confirmed by the following consideration : 
By the equations 
dz a, = 0 and ty = 0 
the pairs of lines are indicated lying respectively harmonically 
with the lines a? = 0 and with the isotropical lines z, = 0. 
And now these two involutions have the pair of rays in common 
of which the equation is obtained by eliminating y between 
Ay Ge yy + Ag Ar Yo = 0 and I + y= 0. 
So the equation 
(ax) az = 0 
represents the orthogonal lines separating a? = 0 harmonically. 
5. If we put 
— 9 
ap Ar be = I » 
then with a view to the equivalence of the symbols a and b, we have 
Gx Jy = ap ag by , 
and 
Gx (gx) = az az (ba). 
But from the identical relation 
Ag bg — ba ag = (ab) (ar) 
follows 
(92) gx = aa (bx) be — (ab) (ax) (bx) « 
The second term of the right member disappearing identically and 
(bx) by =O representing the bisectors of the angles of the lines 
2 = 0, the covariant 
ap az bz 
furnishes two lines having in common withthe lines of a? the axes 
of symmetry. 
At the same time it is evident that the form az ar (bx) does not 
give a new covariant. 
a &4 
