248 
MATHEMATICS: A. B. COBLE 
Proc. N. A. S. 
t we have the pencil of adjoint cubics on the cusp triad of the perspective 
cubic r. 
The form (irx) (dr) (5t) 3 for fixed x and variable r is a pencil of binary 
cubic. This pencil has two linear combinants 9 : a = (irx) (tt'x) (dd 1 ) 
(55') (5t) 2 (5't) 2 and_6 = (irx) (tt'x) (dd') (55')\ The invariants i, j of 
the binary quartic a also are combinants. The invariant / is a sextic 
curve d, the invariant i is b 2 . Hence the discriminant of a factors and the 
two factors b 3 + d and b 3 - d furnish the equations of the cusp locus GC(r) 
and the rational sextic 52(0- We conclude further that there are 12 per- 
spective cubics of S2© with flex points at the meets of b and d. The 
sextics osculate at these points with the flex tangents as common tangents. 
The 12 flex points are the branch points on GC(r) of the function t(r) 
defined by F = 0. Thus a projective (but not a birational) peculiarity 
of GC(r) is that the 12 branch points of one of its series g\ lie on a conic b. 
The parametric line equation of the conic K(r) on which the nodes of 
52(0 determine the nodal parameters of 5i(r) is of degree four in the coef- 
ficients of (HI) . Its symbolic form is (ittV) (ir"'x) (d'r) (55')* (5"5'") 3 
{ (dr) (d"d"') + 2(d"r) {dd'") } = 0. 
With reference to the cubic space curves Ci(r), C 2 (t) the point x deter- 
mines an axis l x of Ci(t) on planes n, r 2 ; r is the third plane of Ci(r) on a 
point y of l x ; and t a plane of C 2 (t) on y. Then to points x on GC(r) there 
correspond axes of C\ on points of C 2 and to the nodes of GC(r) the six 
axes of Ci which are chords of C 2 ; to points x on 52(0 there correspond 
axes of C\ on planes of C2, and to the nodes of 52(0 the ten common axes 
of Ci, G 2 . If x 0 is a node of 52(0 the form (ttx 0 ) (dr) (5t) 3 factors into 
(/ 0 r) (Xo0 • feoO 2 where (q 0 t) 2 is the pair of nodal parameters. The ten 
forms (l 0 r) (\ Q t) will appear later in connection with the symmetroid. 
Other co variants of the (21?) form are easily interpretable with reference 
to C h C 2 . Thus a furnishes the four parameters t of tangents of C 2 which 
meet the axis l x of Ci, and b determines the axes l x of C\ which are in the 
null system of C 2 . 
From the definition of perspective curves the line t f of the perspective 
cubic (irx) (dr) (5t') s will cut the sextic (ttVQ (5t)* (5't)* (dd') in the 
point t = V . On forming the incidence condition of line and point, 
removing the factor (tt') } and setting t' = if, we obtain 
(ttttVO (55') (5t) 2 (5't) 2 (5"ty (d'd") (dr) 
which furnishes the seven contacts 10 1 of the perspective cubic with the 
sextic. This is a form (\\) of general type containing nine absolute con- 
stants which will appear later. 
1 This investigation has been pursued under the auspices of the Carnegie Institution 
of^Washington, D. C. 
2 H. S. White, these Proceedings, 2, 1916 (337). 
3 Meyer, Apolaritdt und Rationale Curven, pp. 320-47. 
