MATHEMATICS: A. B. COBLE 
247 
mations in ll>k(n-k-2) which in general is infinite and discontinuous. 
Associated sets define the same group. The only finite types in this 
doubly infinite (for increasing n and k) series of groups are those de- 
fined by the sets mentioned above or by their associated sets. Each 
Gn,k contains Gn-\,k as a subgroup. 
The Cremona transformations in Sk under which the congruence 
of sets Pn^ occurs will have a certain effect upon the spreads in Sk. 
If the order of the spread and its multipKcity at each point of Pn^ 
be taken as variables,^ this effect is expressed by a linear transforma- 
tion on n -\- \ variables. Thus there is determined a group of hnear 
transformations riik in O n with integer coefficients which is isomorphic 
with the Cremona Gn^k. In S'n, the gn^k has an invariant S'n-i within 
which it determines an isomorphic g^n,k of linear transformations whose 
coefficients are rational numbers. The importance of the finite types 
in these series of groups would seem to indicate that interesting appli- 
cations of the infinite types may be expected. 
A definite appKcation of the foregoing theory to the determination of 
the Hues on a general cubic surface will be made.^ As has been 
mentioned earlier, cubic curves on P^^ map S2 upon a general cubic 
surface which appears in Cremona's hexahedral form, O. The rational 
invariants of Pe^ are rational invariants of the surface after the ad- 
junction of the irrationality which isolates a set of six skew lines of a 
double six. Such invariants determine, and are determined by, the 
invariant spreads of fti in 24. If it be required further that the spreads 
be invariant under the extended group, Gq,2 of order 51840 in 24 they 
determine the invariants of O itself. In this way not only the in- 
variants but also the Hnear covariants of are calculated and iden- 
tified with those of K^. Now being given, its invariants and linear 
covariants are known. The invariants are the known quantities in the 
form-problem of ^6,2- The solution of this form-problem furnishes a 
point P in 24, the map of P^^. From P^^ as indicated earher the lines 
of as well as the linear covariants of are derived. The linear 
transformation obtained from the identification of the linear covari- 
ants of O and transforms the known lines of into the required lines 
of K^. All of the processes outlined here are rational except the solu- 
tion of the form-problem of ^6,2. This may be dismissed as a problem 
in the theory of functions or it may by means of an accessory irration- 
ality be reduced to the form-problem which occurs in the trisection of 
the periods of the hyperelliptic functions of genus two. The ultimate 
importance of this latter problem has been pointed out by Klein. ^ It 
would be more desirable however to utilize directly the theta-modular 
