( 789 ) 
The maximum value of a , for which p breaks up into at least n 
different a-groups we shall call the n-partite width of dispei'sion of p, 
and shall represent it by (ft). Clearly 6 „(p) is < d (p). 
For p exists furthermore a series of increasing positive integers 
n 1 (p'), n,(n), .... in such a way that <f»(p) for n between 
nk~ i (p) and n k (p) is equal to d„ k o4p)- This quantity we call 
the £ th width of dispersion of p and as such we represent it by 
d(*)(p). 
We now assert that it is always possible to break up p into m x 
perfect sets of pieces p 15 .... p Bll so as to have d(PA)^ d Wl (p) and 
«(PA 15 PO^ d Ml (P). 
Let namely be cf Ml (p) = rfW(p); we can then obtain the required 
number m 1 by composing each p* of a certain number of dW(p)- 
groups belonging to a same : O(p)-group. We are then also sure 
of having satisfied the condition «(pa x , Pa s )^ ^(p)- 
Fartheron we can place the rf( fc )(p)-groups of a same dC*-0(p)-group 
into such a row that the distance between two consecutive ones is 
equal to cf( fc )(u). If we take care that each p* consists of a non¬ 
in terrupted segment of such a row, then the condition <%*) ^ d« t (p) 
is also satisfied. 
Let ns now break up in the same way each p* into ra, perfect 
sets of pieces p A1 , . . . . phm, in such a way that d(p M ) ^ <Mpa) and 
, p Wj ) ;> d w fnh), and let us continue this process indefinitely. 
If 1 then we represent by r, an arbitrary row of v indices, then 
we shall always find 
^> Fv ^^d« l+ ^:...+*,-|-i-v(p) • * • • ( A ) 
As p is a perfect set of pieces, the width of dispersion d(p F J 
can converge to zero only for indefinite increase of v, out of the 
formula (A) follows, however, that for indefinite increase of i> that 
convergency to zero always takes place and, indeed, uniformly for 
all v l]l elements of decomposition together. 
At the same time the separating bound of every two pieces lying in 
one and the same v* element of decomposition converges uniformly 
to zero; so these elements of decomposition converge themselves 
uniformly each to a single piece. 
If finallv a variable pair of pieces of p is given, then their distance 
can converge to zero only when the order of the smallest element 
of decomposition, in which both are contained, increases mdefimtely. 
The simplest mode in which this process of decomposition can be 
