.246 
MATHEMATICS: A. B. COBLE 
Proc. N. A. S. 
SlM, 5,(0 
Si(r), S 2 (t). 
Two sextics in a row of the array will be called paired sextics; two in a 
column, counter sextics; and the other pairs, diagonal sextics. If any one 
of these sextics be given, its 1% spread out on a space cubic will determine 
the other space cubic and thereby the entire set of four. The nodal 
parameters of the paired sextics in the upper row are those of the ten 
common chords of Ci, C 2 ; in the lower row, those of the ten common axes 
of Ci, C 2 . The equations of the sextics are 
(a a'Y (at) (a't) (flr) 3 (a'r) 3 = 0, (A A') 2 {At) (A'r) (A*)'* (A'*) 3 = 0, 
(A A') 9 (At) (A't) (At) 3 (A't) 3 = 0, (a a') 2 (or) (a'r) (at)* (a't)* = 0. 
Here the coefficients of the quadratics in t or r furnish three line sections of 
the respective sextic. The significance of the quadratic parameter appears 
in 6. 
4. Two Birationally Related Quartic Surfaces. — The two nets Qu Qi of 
point quadrics on Ci, C 2 , respectively, are apolar to a web of quadric en- 
velopes Q; similarly the nets Qu Q 2 are apolar to a web of point quadrics, 
Q. The jacobians, J, J, of these respective webs are quartic envelope or 
surface, respectively ; the first on the ten common chords, the second on 
the ten common axes of G, C 2 . If we map by means of the web Q its 
space upon another space, the jacobian /, the locus of nodes of quadrics in 
Q, is mapped upon a surface 2 of order 16 and class 4, the Cayley symme- 
troid quartic envelope with ten tropes. The two cubic curves are mapped 
upon two paired rational space sextics Ri(t), R 2 (t) which are conjugate to 
the paired rational plane sextics 5i(r), 52(0, respectively, i.e., plane sec- 
tions of the space sextic are apolar to line sections of the Conjugate plane 
sextic. The symmetroid 2 is the locus of planes which cut the sextic R- t 
in catalectic sections. Similarly the jacobian J counter to J is mapped by 
the web Q upon a point symmetroid 2 counter to 2, and Ci, C 2 upon 
rational space sextics Ri(t), R 2 (t), counter to Ri(t), R 2 (t), respectively, 
and conjugate to 5i(r), S 2 (t), respectively. 
5. References'. — Meyer 3 has discussed the relation of J to the sextic 
52 (0 and mentions the occurrence of counter sextics. Conner 4 considers 
the mapping from J to 2 and its connection with the paired rational sextics. 
The above introduction of the tetrad of rational sextics as defined by the 
sextics F, F of genus 4 is novel. Schottky, 5 beginning with the abelian 
theta functions of genus 4, derives a set of ten points in space which are 
the nodes of a quartic surface and merely states a characteristic property 
of this surface by which it can be identified with 2. The writer 6 has 
shown that 2 can be transformed by regular Cremona transformation 
into only a finite number of projectively distinct symmetroids. These 
classes permute under the group (mod. 2) of integer transformations of the 
periods of the functions of genus 4. The analogous result for the plane 
