MATHEMATICS: A. B. COBLE 
245 
POINT SETS AND ALLIED CREMONA GROUPS 
By Arthur B. Coble 
DEPARTMENT OF MATHEMATICS. JOHNS HOPKINS UNIVERSITY 
Presented to the Academy, January 18, 1915 
It is my purpose to develop, in a series of papers of which the first 
has been submitted to the Transactions of the American Mathematical 
Society, some aspects of the theory of an ordered set of n discrete 
points in a hnear projective space, Sk, of k dimensions.^ Such a set, 
denoted by Pn^, defines projectively an 'associated' set Q 
the two sets being mutually related. The study of this type of asso- 
ciation, begun by Rosanes and Sturm,^ is continued here. li n = 
2k -\- 2 the associated sets can lie in the same Sk, and in particular 
they may coincide in order forming thus a 'self-associated' set. In 
addition to projective constructions for associated and self-associated 
sets, certain irrational conditions for such types of association which 
seem to be novel are given. 
The invariants of a set are formed from the determinants of the 
matrix of the coordinates of the points. An invariant is required to be 
homogeneous and of the same degree in the coordinates of each point 
and to be unaltered under permutations of the points. It appears that 
associated sets have proportional invariants. This theory of invari- 
ants is the natural extension of the theory of binary forms if the form 
be regarded as the definition of a point set Pn^. Complete systems 
for and Pi are derived and the accompanying algebra is utiHzed 
to obtain explicit equations not only for the cubic surface mapped from 
the plane by cubic curves on Pi but also for the tritangent planes and 
the Hnes of the surface. 
By a certain construction the ordered set Pn* is mapped upon a point 
P of a space 2A;(n-A;-2). The Cremona transformation from one map 
P to another map P^ of the same set is discussed and eventually the 
construction is simpKfied so that this transformation is linear. The 
n\ permutations of the points of Pn* lead to n\ points P in 2 which are 
conjugate under a Cremona group Gn\ in 2. When k = \, Gn\ is 
the 'cross-ratio group' of Moore.^ A set of generators of Gn\ is exhib- 
ited. The invariant spreads of Gn\ are obtained from the invariants 
of Pnf^. The linear system of spreads of lowest order invariant under 
Gn\ is derived from the irrational invariants of PrJ" of lowest weight. 
Associated sets determine the same group. 
The form-problem of the determined by the associated P5I and 
Pi leads to a solution of the quintic equation ; that of the Ge! determined 
