(791 ) 
the finite continuous groups of Lie, the geometric type of order £ 
admits of groups of continuous one-one transformations, which possess 
likewise the geometric type of order £. 
In order to construct such groups we start from a decomposition 
according to $ 3 of the set p into m x “parts of the first order” 
Pm x , of each of these parts of the first order into m 2 “parts 
of the second order” phi, pm, pas, • ■ • Pkw, , etc. 
The parts of the first order we submit to an arbitrary transitive 
substitution group of m x elements, of which we represent the order 
by p x , and which we represent itself by g x . 
After this we submit the parts of the second order to a transitive 
substitution group -g 2 of m x m 2 elements which possesses the parts of 
the first order as systems of imprimitivity and g x as substitution 
group of those systems into each other. We can then represent the 
order of g 2 by p x p 2 . 
The simplest way to construct such a group g 2 , is to choose it 
as the direct product of g x and a substitution group y a , which of 
the parts of the second order leaves the first index unchanged and 
transforms the second index according to a single transitive substi¬ 
tution group of m 2 elements. 
We then submit the parts of the third order to a transitive sub¬ 
stitution group g t of m x m 2 m i elements which possesses the parts of 
the second order as systems of imprimitivity and g t as substitution 
group of those systems into each other. We can represent the order 
of 9 * b 7 Pi V . V »• 
In this way we construct a fundamental series of substitution 
groups g v g„g t ,"- 
Let Tj be an arbitrary substitution of g x ; r t a substitution of g 2 
having on the first index of the parts of the second order the same 
influence as t, ; t 3 a substitution of g, having on the first two indices 
of the parts of the third order the same influence as r,; and so on. 
The whole of the substitutions r„ then determines a substitution 
of the different fundamental series of indices into each other, in other 
words a transformation r of the pieces of p into each other. 
This transformation is in the first place a one-one transformation; 
for, two different pieces of p lie in two different parts of a certain, 
e.g. of the r* h order, and these are transformed by r into again two 
different parts of the r th order. 
If fartheron S x , S„ S 3 ,.. . is a fundamental series of pieces, pos¬ 
sessing So, as its only limiting piece, then, if *(») is the lowest possible 
order with the property that S n and S* lie in different parts of that 
order, X(n) must increase indefinitely with n. 
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