( 793 ) 
If now we adjoin to each substitution group g n a finite group 
g' n of continuous one-one transformations of p as a set of pieces in 
itself, tranforming of the pieces of p the first n indices according to 
g n , but leaving unchanged all their other indices, then the funda¬ 
mental series of the groups g\ , g \, g \,... converges unifoimly to 
the group g. 
The set whose elements are the groups g of the geometric type of 
order C constructable in the indicated manner possesses the cardinal 
number of the continuum. For, already the set of those series 
m lt m,, m t ,... , which consist of prime numbers, possesses this 
cardinal number, and any two different series of this set give rise 
to different groups g. 
We can sum up the preceding as follows: 
Theorem 5. The geometric type of order $ allows of an infinite 
number of groups consisting of a geometric type of order £ of con¬ 
tinuous one-one transformations and being uniformly approximated 
by a fundamental series of groups consisting each of a finite number 
of continuous one-one transformations. 
If in particular we consider those groups g for which each g n is 
chosen in the way described at the commencement of this § as the 
direct product of g n — 1 and a group y„, we can formulate in par¬ 
ticular : 
Theorem 6. The geometric of older £ allows of an infinite number 
of groups consisting of a geometric type of order £ of continuous 
one-one transformations and being uniformly convergent diredt pro¬ 
ducts each of a fundamental series of finite groups of continuous 
one-one transformations. 
* 5 . 
The sham-addition in the geometric type of order S. 
Let us choose the factor groups indicated in theorem 6 as simply 
as possible, namely g 1 as the group of cyclic displacements corre¬ 
sponding to a certain cyclic arrangement of the first indices, and 
likewise each y„ as the group of cyclic displacements corresponding 
to a certain cyclic arrangement of the n th indices; g is then com¬ 
mutative, and transitive in such a way that a transformation of g 
is determined uniformly by the position which it gives to one of 
the elements of p. 
Let us further choose an arbitrary piece of fi as piece zero. Let 
us represent this piece by S 0 , and the transformation, which trans¬ 
fers S 0 into *Sa and is thereby determined, by That the 
