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MATHEMATICS: A. B. COBLE 
Proc. N. A. S. 
forms of each other under the quintic Cremona transformation T but on 
the two the role of the t and r triads with reference to the rational sextics 
and conies K is reversed. It may be shown that the projective peculiarity 
(equivalent to four conditions) of our birationally general sextics GC(t) 
and GC(t) is that their r triads envelop a conic K(t). This same property 
belongs to the t- triads after transformation of GC(t) by T (or GC(t) by 
the corresponding T) into a sextic in the plane S y . 
We have mentioned in I 7 certain intersectional properties of the conic 
b, the rational sextic S2W , and the cusp locus GC(r). Analogous results 
for the conic K(t), the sextic S2W , and the locus GC{r) are as follows. 
The conic K(t) meets 52(0 in 12 points whose parameters t on 52(0 are 
the branch points of the function r(t) determined by F = 0, and whose 
parameters r on K(t) are the branch point values of this function. The 
12 residual function values r furnish 12 common tangents of K(r) and 
GC(r), i.e., the tangents to GC{r) at the 12 coincidences of gi 3 (r) upon it 
touch K(t). The remaining common tangents of K(t) and GC{r) are 
double tangents of GC(t) whose parameters on K{r) are the branch points 
of the function t(r) in F = 0. The two further tangents to K(t) from 
the contacts with GC{r) of each double tangent meet in a point which is 
a coincidence of gi 3 (0 on GC{r). 
11. The reciprocity between the forms F, F; algebraic discussion. — We 
denote by A the determinant of the coefficients (without binomial factors) 
of the form F ; the coefficients of F are then, to within numerical factors, 
three-row minors of A. The reciprocity between the forms F, F is then 
brought out by the fact that the covariant F formed for F as a ground form 
is again F.A 2 / 3 4 . The invariant (aA) z («A) 3 is 4 A. This reciprocity 
fails when A vanishes. Then the four cubics in t furnished by F are 
linearly dependent. When expressed in terms of three the coefficients 
in F are three cubics in r, and the form F = 0 can be interpreted as the 
incidence condition of point r on a rational plane cubic curve C\ with line 
t of a rational plane cubic envelope C 2 . In this case the form F factors 
into two cubics (At) 3 . (A*) 3 , the conjugate cubics of the rational curves 
Ci, C 2 . Now all the (usually non-vanishing) odd transvectants of the 
double form F will vanish whence in the general case they contain the 
factor A and are of the second degree for F. From their degree and orders 
they can be identified at once with 2 / 9 (6r) 4 (fit) i = (aa') (aa') (ar) 2 
(a'r) 2 (a/) 2 («'0 2 J 8 5 = (aa') 3 (cm') 3 ; 4 A ( 7 *) 4 = (™') 5 (««0 («0 2 (<*'0 2 ; 
Vs(ct) 4 = (aa') z (aa') (ar) 2 (a'r) 2 . With respect to the original space 
cubics Ci t C 2 these forms are interpreted as follows. The form (6r) 4 (/3*) 4 
vanishes when tangent r of C\ meets tangent t of C%\ 5 = 0 if the 
null systems of G, C 2 are apolar; (7O 4 or (cr) 4 vanishes if the tangent / 
of C 2 or the tangent r of C\ is a line of the null system of the other curve. 
The reciprocity between the dual forms F, F is due algebraically to 
