MATHEMATICS: A. B. COBLE 
575 
POINT SETS AND CREMONA GROUPS. PART HI 
By Arthur A. Coble 
DEPARTMENT OF MATHEMATICS, JOHNS HOPKINS UNIVERSITY 
Received by the Academy, August 23, 1916 
In Part of this series projectively distinct sets Pj^ of n points in 
Sk were mapped upon points of a space ^kin-k-2) and a certain Cre- 
mona group Gn! in 2 was obtained by permutation of the points of 
the set. In Part 11^ the Gn appeared as merely a subgroup of a more 
important group Gn, a in S k{n-k-2) which also is defined by PrJ'. In 
particular the Ge, in 2:4 attached to Pe^ is a subgroup of the G%, 2 in ^4 
which has the order 51840 and is isomorphic with the group of the 
lines on a cubic surface. 
The purpose of this Part III is to utilize the Ge, 2 in the problem of 
determining the lines of a cubic surface It appears that there is a 
one-to-one correspondence between the invariants of O and the invari- 
ant spreads of Ge, 2 in 24. The lines of O can be rationally expressed 
in terms of a solution of the form problem of Ge, 2 by means of a typical 
representation of in the hexahedral form with the aid of the linear 
CO variants of C^. In order to solve the form problem of Ge, 2 the sim- 
plest linear system of irrational invariants of is employed. This 
system is of dimension 9 and the members appear in 24 as quintic 
spreads. Under the invariant subgroup Pe, 2 of Ge, 2 of index two this 
linear system separates into two skew linear systems each of dimen- 
sion 4 with the important property that the members of the two sys- 
tems are permuted under the operations of Ge, 2 precisely as the points 
and ^s's of a linear space 5*4 are permuted under the elements of a cor- 
relation group in 5*4 whose collineation subgroup is the Burkhardt 
group G25920 in 6*4. The form problem of Ge, 2 can then be solved in 
terms of a solution of the form problem of G25920 by using the point in- 
variants of G25920 and in addition five invariants of G25920 linear in the 
6*3 coordinates and of degrees 1, 7, 9, 13, 15 in the point coordinates. 
The method for solving the form problem of G25920 is suggested by 
the properties of the normal hyperelliptic surface M^"^ of grade 3 in 
6*8 obtained parametrically by using 9 linearly independent theta func- 
tions of the third order and zero characteristic. The M^"^ admits a 
collineation group G2.^i which contains 81 involutions. If / is one of 
these involutions the fixed Sz and fixed of / meet M^^^ in 6 and 10 
points respectively. The M2^^ is projected from the fixed upon the 
fixed 5.3 into a doubly covered Weddle surface and from the fixed Sz 
upon the fixed 5*4 into a doubly covered 2 -way which has a node 
