(m > 
Bo by the transformation r the fundamental series passes into a 
new fundamental series having as its only limiting piece the piece into 
which passes by t. 
As a set of pieces n is thus continuously transformed by r. 
Let r\, r' f , r’ 3 ,. .. be a series of substitutions satisfying the same 
conditions as the series r lt r 2 , r„ .. . If then r 1 x\ = r l ; *■?■ = * ,; 
etc., then the series r\, t'\, t"„ .... likewise satisfies the same conditions. 
If furthermore t' and r" are defined analogously to r, then r x' is 
equal to t". 
So the transformations satisfying the conditions put for r form a 
group , which we shall represent by g. 
To investigate the geometric type of order .of this group, we 
decompose in the way indicated in § 3 a perfect set of pieces p into 
p x parts of the first order p t , p*, • • • •, Q Pl ; each of these into p t 
parts of the second order pAi, Qh 2*--and so on. 
The p 1 substitutions of g x we bring into a one-one correspondence 
to the parts of the first order of p. Then the p x p 2 substitutions of 
g. t into such a one-one correspondence to the parts of the second 
order of <>, that, if a substitution of g % and a substitution of g x have 
the same influence on the first index of the parts of the second order 
of fi, the part of the second order of p corresponding to the former 
lies in the part of the first order of p corresponding to the latter. 
In like manner we bring the p l p t p 2 substitutions of g t into such 
a one-one correspondence to the parts of the third order of 9, that, 
if a substitution of g t and a substitution of g 2 have the same influence 
on the first two indices of the parts of the third order of f 1, the 
part of the third order of q corresponding to the former lies in the 
part of the second order of p corresponding to the latter; and so on. 
The parts of p corresponding to a series r l5 t 2 , t 3 ,... then converge 
to a single piece of q, which we let answer to the transformation 
r deduced from the series. Then also inversely to each piece of Q 
answers a transformation r, and the correspondenceattained in this 
manner is a one-one correspondence. 
Farthermore two transformations t and x converge to each other 
then and only then, when their generating series ,r t , r s ,-and 
r\ . r' s , V,,_have an indefinitely increasing commencing segment 
in common, in other words when the corresponding pieces of Q 
converge to each other. So the correspondence between the trans¬ 
formations r and the pieces of p is continuous. 
The transformations t, in other words the transformations of the 
group g , have thus been brought into a continuous one-one correspon¬ 
dence to the pieces of p, so that g possesses the geometric type of order 
