246 
MATHEMATICS: A. B. COBLE 
by the associated and P^^ and by the self-associated Pg^ to a solu- 
tion of the sextic equation. These equations have been discussed from 
this point of view in earlier papers of the writer.^ The principal novelty 
here is the use of a connection between the self-associated P^^ and the 
theta modular functions of genus two to solve the related form-prob- 
lem. In general one may say that the theory of the point set furnishes 
a common algebraic background not only for the general equations 
of degree n but also, as will appear later, for such particular equations 
as are involved in the determination of the lines on a cubic surface and 
of the double tangents of a plane quartic curve. 
By introducing certain conventions it is possible to establish a mutual 
order for the two sets of singular points of a Cremona transformation 
in the plane. Since ordinary corresponding point pairs also are mutu- 
ally ordered the addition of any nimiber of ordinary pairs to the sets of 
singular points leads to a pair of sets which are said to be ^congruent' 
under Cremona transformation. The conditions under which congru- 
ence can occur are fundamentally important for effective application 
of the Cremona transformation to geometry or analysis. These con- 
ditions are derived up to a certain natural limit. Since two sets con- 
gruent to a third are congruent to each other the projectively distinct 
sets in the plane which are congruent in any order with a given set 
Pii^ will be mapped by an aggregate of distinct points in S such that 
the aggregate is determined by any one of its points. If the type 
of transformation under which the congruence occurs and if the order 
be given the set is uniquely determined so that the transition from 
one point in 2 to another is effected by a Cremona transformation. 
In other words the sets in^ congruent in any order with a given in^ 
are mapped in 2 by points conjugate under an 'extended' Cremona 
group Gn,2 which contains Gn\ as a subgroup. For n = 6, 7, 8 the 
order of Gn,2 is 61.72, 71.288, 81.2M35 respectively but for larger values 
of n the Gn,2 is infinite and discontinuous. The finite groups are 
isomorphic respectively with those of the lines on a cubic surface, the 
bitangents of a plane quartic curve, and the tritangent planes of a 
space sextic of genus four on a quadric cone. 
Let us call a Cremona transformation in Sn which can be expressed 
as a product of the particular transformations obtained by inverting 
the variables a 'regular' transformation. Obviously the regular trans- 
formations are the elements of a 'regular group.' Then the definition 
of congruence of point sets in S2 as given above can be generalized to 
include congruence under regular transformations in Sn. Again the 
set Pn^ defines as above an extended group Gn,k of Cremona transfer- 
