( 489 ) 
It is clear that ay (ax) (be) represents the lines at right angles to 
abe br: == 0s 
6. To two quadratic forms a? and „2 belong the simultaneous 
invariants 
(af, at=f2, (Dar. 
As is known the first disappears when the two pairs of lines 
separate each other harmonically. 
Under the condition (af) ay = 0 the lines determined by (af) ax fz=0 
are perpendicular to one another. 
These right lines being the double rays of the involution 
a2?+Af2=0, the equation (af) a= 9 indicates that the pair of 
lines a2 =0 and f? =0 have common axes of symmetry. 
This is confirmed by the following consideration. We have 
(af) af = (as — 4%) fir — 411 (foo — Soa) « 
If fi, = 0, the invariant disappears when at the same time a,;=0 
or foo = foo, i.e. when the two pairs of rays have common bisectors 
or when one pair of rays consists of the isotropical lines. 
From the expression found above for the tangent of the angles 
of a pair of lines follows readily that the invariant 
PL, 9, — UD e, b, 
disappears, when these pairs of lines can be brought to coincidence 
by the rotation of one of them. 
1. When the equations 
a SS Ugg TF + 2 ay 2] Xo + dz =, 
(fe)? = Jog #2? — 2 fir mvo + fon 5 = 0 
have a root zj : zy in common, one of the lines a? = 0 is at right angles 
to a line of #2 = 0. So the resultant of these equations must furnish 
a simultaneous invariant. 
36 
Proceedings Royal Acad. Amsterdam, Vol. IL 
