Vol.. 7, 1921 
MATHEMATICS: A. B. COBLE 
247 
rational sextic involves a subgroup of the group (mod. 2) of the periods of 
the functions of genus 5. This indicates a connection (which we seek) 
of the functions of genus 4 and those of genus 5. Proceeding the other 
way Wirtinger 7 obtains the plane sextic of genus 4 as the locus of vertices 
of diagonal triangles of a linear series g\ upon a ternary quartic (p = 3). 
This transition will be discussed later. 
6. The Covariant Conic R(r) of the Rational Plane Sextic S2(t). — From 
the existence in the net Qi of a quadratic system of cones we conclude 
that the rational sextic 52(0 has a covariant conic K(t) such that the ten 
nodes of S 2 (0 determine upon K(t) the ten pairs of nodal parameters of 
the sextic Si(t) paired with the given sextic 52(0- This theorem furnishes 
the bond between ten nodes as a ternary figure and ten nodes as a binary 
figure on the rational curve. The equation of the sextic in Darboux 
coordinates referred to the norm conic K(r) is precisely that given in 3. 
7. The Perspective Cubics of S2(0- The form (2I1). We denote by 
the symbol Cki,%l. '.'.".'.'.) an algebraic form of order i\ in the variables of an 
Ski, of order i 2 in the variables of an Sfo, etc. Unless explicitly restricted 
these sets of variables are digredient. Thus F =(ar) 3 (at) z is a form 
(if). By polarizing F into (an) (ar 2 ) (ar) (at) s and replacing the pair 
of parameters n, ri by the point % which they determine in the plane of 
K(r) we obtain the (HI) form 
(ttx) (dr) (8t) 3 
a general form of the orders indicated with nine absolute constants. For 
given r this form determines a rational cubic envelope perspective 8 to the 
sextic 52(0, i.e., line / of the cubic is on point t of the sextic. The sextic 
is the locus of the meets of corresponding lines of any two of the 00 1 per- 
spective cubics, and it has the equation (inr'£) (dd') (8t) 3 (8 f f) z = 0. 
Conversely given the sextic the family of perspective cubics is determined. 
Each cubic r has three cusps whose parameters are given by (xti-V) (dr) 
(d'r) (d"r) (55') 3 (5"0 3 = 0. This is F = (Ar) 3 (AO 3 whence the cusp 
locus, GC(t), is birationally general and of genus 4. The equation of the 
cusp locus is the determinant of the coefficients of (ttx) (tt'x) (dr) (d'r) 
(88 ') 2 (50 (8't) regarded as a form bi-quadratic in r, t. Thus GC(r) is a 
sextic whose six nodes are the points for which the first minors of the above 
determinant vanish and these first minors furnish the nine linearly in- 
dependent quartic adjoints of GC(r). 
The curve of genus 4 has two special series g?, residual with respect to 
each other in the canonical series. These appear in the normal form as 
the triads on the two sets of generators of the quadric containing the 
sextic. One of these series on GC(r) is the triads of cusps of perspective 
cubics of 52(0 • The web of adjoint cubics of GC(r) is furnished by the form 
(ttx) (t'x) (tt"*) (dd') (88 r ) (8'8 W ) (88"Y (8't) (d'r) = 0, 
t and r being variable with the cubic of the web. For fixed r and variable 
