MATHEMATICS: A. B. COBLE 
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points of Pn a Cremona group Gn\ which for = 5, 6 led to new ver- 
sions of the solutions of the quintic and sextic equations. 
In Part II the novel idea of the congruence of point sets is introduced. 
This notion can be defined at once for a space S2. Two mutually ordered 
point sets Pi and P!^ are congruent under a ternary Cremona trans- 
formation C with p ^n fundamental points {F = points) if p of the pairs 
comprise the F-points of C and and if the remaining n — p pairs of the 
sets are pairs of ordinary corresponding points of C. Thus the number 
of types of congruence depends upon the number of types of C and the 
number of ways of ordering the two sets. There is a natural pairing of 
the p F-points of C with the p F-points of and this is utilized to state 
in quite compact form the conditions for congruence up to w = 9. These 
conditions imply when p = n a, construction for the two sets of F- 
points and when p < w a construction for the transformation C. The 
importance of the notion of congruence is due to the fact that two sets 
congruent in some order to a third are congruent in some order to each 
other. Thus if all the sets P!^ congruent in some order to a given set 
Pn be mapped upon points of S2(n-4) they form a conjugate set of 
points under the operations of a Cremona group G„,2 in 22(n-4) which 
contains the Gn\ of Part I as a subgroup. 
This definition of congruence cannot be extended immediately to 
sets in Sk. Let us first define a regular Cremona transformation C in 
Sk to be one which can be generated as a product of pro jectivi ties and 
of inversions of the variables. The regular transformations constitute 
a regular Cremona group in Sk. They are determined by their F-points 
precisely as in 5*2- It is now possible to define as above congruence of 
sets Pn and Pn under regular Cremona transformation in Sk and the 
group Gn,k in l^k{n-k-2)' 
The effect of a regular transformation on spreads in Sk of order 
with multiple points of orders Xi, . . . Xn at the points of Pi is repre- 
sented by a linear transformation on these variables with integral coeffi- 
cients. In this way a group gn,k is derived which is isomorphic with 
Gn,k. This group reveals the striking analogy between the general 
transformation in S2 and the regular transformation in Sk. 
All sets Pn congruent to a set PI upon an elliptic norm-curve in 
Sk can be projected upon the same curve. The transition from PI 
to Pn is then effected by a linear transformation on the elliptic parame- 
ters Ui, . . . , with rational coefficients. In this way a group 
en,k is derived which also is isomorphic with Gn.k. The group is useful in 
determining the cases in which congruence implies projectivity. 
The close relation of associated sets F„ and is again apparent 
