( 491 ) 
Setting aside the forms (ab) a and (ab)? which disappear identi- 
cally, we have the invariants 
2 
(ab)? ay = 2 (aso Uig an a1 rr a, + 493 daj) ’ 
Bd HI HIA taf, 
From the identity (ab)? + a? = aa by evidently follows 
aa ay by = (ab)? ay + a8 3 
For «39 = 0 and G3 = 0 we have 
a? = 3(a2 + at) and (ab)? ap = — 2(a2 +4)» 
so 
2 a} + 3 (ab)? ay = 2 (a2, + 8 azo arg + 3 agg den + a) =O. 
Reciprocally the disappearing of this invariant indicates that two 
lines of a3 = 0 are at right angles to each other. For, if by a rotation 
of the axes of coordinates a’ is transformed into 
Sans So + 3 ys § 2 + 83 
which implies that one of the lines is represented by £3 = 0, then the 
angular coefficients of the remaining lines are connected by the 
relation mg m3 = 3 @,. The above named invariant being transformed 
into 8 ax +1, its disappearing produces the relation mg m3 + 1 = 0, 
by which two perpendicular lines are indicated. 
9. The comitant 
a, a = 0 
determines the polar of a’ with respect to the line yi: y= 2: #2, 
or, what comes to the same thing, the double lines of the cubic 
involution of which (#y)=0 is a threefold ray and a? = 0 forms 
a group. 
For the double rays of the involution 
36* 
