Vol.. 7, 1921 
MATHEMATICS: A. B. COBLE 
335 
line-sixes. The mapping of the surface from the planes S x and S y is 
given by the above systems of cubics. 
9. The form (\ { \ \). — In the space figure just described we insert a quadric 
Q with generator o t, r. The point coordinates z can be replaced by bilinear 
expressions in t, r and the form (f f f) becomes a form (pr) (irt) (roc) (sy) 
general of its type, with thirteen absolute constants corresponding to the 
four absolute constants of the above cubic surface and the nine additional 
constants for the inserted quadric. The quadric meets the cubic surface 
in a normal curve of genus 4 so that we have in space the figure of the 
normal sextic and a particular one of the oo * cu bic surfaces through it. 
In the planes S x , S y we have projectively general (with thirteen absolute 
constants) sextics of genus 4 with nodes at p { and g t -, respectively, trans- 
forms of each other under T. The canonical adjoints of these sextics, 
with isolated gi v s, are obtained by substituting for u in 8 the proper 
bilinear expressions in t, r ; while a similar substitution for z in the equation 
of the cubic surface gives the equation on the quadric of the normal curve. 
We may therefore regard the general form ({ \ \ f) as a definition of the 
projectively general plane sextic of genus 4. 
10. The forms F,F on the conic K(r). Counter sextics. — We mark the 
conic K(r) on the plane S x and plot with reference to it the two counter 
sextics 52(0 and S 2 (t). For each there is a cusp locus of perspective cubics 
GC(t) and GC(r). Similarly we mark on a plane S y the conic K(t) and 
plot with reference to it the two counter sec tics 5i(r), 5i(r) which are 
paired with the above counter sextics, and which have for cusp loci of 
perspective cubics the sextic curves GC(t), GC(t), respectively. 
We now consider the form F = (at) 3 (ar) 3 = 0 where r is a tangent 
of K(r).. For variable t we have 00 1 triangles circumscribed about K(r) 
whose vertices run over a sextic curve. If t determines n, r 2 , r 3 then the 
point Tit T 2, of this sextic curve is birationally related to the solution t, 
t 3 of F — 0. Hence the sextic curve is birationally equivalent to 
GC(t) and as a result of the algebraic discussion of the next two sec- 
tions we prove that this sextic curve is actually GC(r). This amounts 
to the effective elimination of t from (an) 3 (at) 3 = 0, (ar 2 ) 3 (at) 3 = 0 
and the separation of the factor (tit 2 ) 3 . Hence we have on the sextic 
GC(t) two gi v s such that the 00 1 ^-triads are cusp triangles of perspective 
cubics of 5 2 (0 and the 00 1 r-triads are triangles circumscribed to K(r) for 
which the intersection of the lines joining each vertex to the contact of 
the opposite side is the point t of 5 2 (0- Of course a similar statement is 
true of any one of the four cusp loci. 
If we polarize the form F into (ar) (an) (ar 2 ) (at) (ah) (ah) and replace 
the pairs n, r 2 ; t h h by points x; y referred in Darboux coordinates to the 
conies K(t) ; K(t) in Sx; Sy, respectively, we have a form (irx) (dr) (dt) (py) 
of the type discussed in 9. Hence the sextics GC(r) and GC(t) are trans- 
