246 
MATHEMATICS: A. B. COBLE 
in that the groups Gn,k and Gn, n-k-2 are identical while the groups 
gn, k and gn, n-k-2, as well as the groups e„, ;b and n-k-2, are linear 
transforms of each other. 
The group G„. a is in general infinite and discontinuous. The only 
finite types are the Gq,2, the G7,2 = Gi,z and the G%,2 =Gb.4' These are 
identified with the well known groups of respectively the lines on a cubic 
surface, the bitangents of a quartic, and the tritangent planes of a sextic 
of genus 4 on a quadric cone. For these cases Gn,k is in immediate 
algebraic relation to the corresponding geometric configuration. This 
advantage is used in the case of P7 to determine the simplest system of 
irrational invariants of the point set which are invariants of the allied 
quartic as well. A similar method will be employed in Part III to 
handle the Pg 2<nd the allied cubic surface. The first cases of infinite 
order, the G9, 2 = G9, 5 and the Gg, 3 can be adequately discussed by means 
of the isomorphic group en,k and their structure has been determined. 
Some interesting by-products are obtained. By means of gn,k a 
determination of all types of regular Cremona transformations with a 
single symmetrical set of P-points is made. The new types thus found 
are a transformation in Si of order 49 with 8 P-points of order 30 and a 
transformation in 5*2 of order (2k^ — l) with 2 + 1) P-points of order 
2k (^ — 1). Also the discussion of ^9, 2 leads to a determination of the 
infinite number of types of ternary Cremona transformations with 9 
or fewer P-points in terms of 8 independent integers. Theorems such 
as the following: — A pencil of plane cubic curves can he transformed by 
ternary Cremona transformation into only 960 projectively distinct pencils 
of cubics — are proved for special sets Pg. Similar facts can be 
derived for Pg. Furthermore the general methods developed can be 
employed in the problem of determining the finite groups of regular 
transformations in Sk. For if a point P in 2 k(n-Jc-2) is fixed under a 
certain subgroup of Gn,k the corresponding set P^ in Sk defines a regular 
Cremona group in Sk isomorphic with the given subgroup. 
1 These Proceedings, 1, 245 (1915) and Trans. Amer. Math. Soc, 16, 155 (1915). This 
investigation has been carried out under the auspices of the Carnegie Institution of Wash- 
ington, D. C. 
