203 
eristics, that of the vertices and that of the limiting »—-1-dimensional 
polytopes, for half a characteristic, or — what comes to the same — 
by counting each of the two extreme characteristics and each of the 
two halves of the remaining intermediate characteristics for one. So 
the scale relating to our space passes into 
1 2 
=r DN 
4 4 4 
| | | 
E P bie et R 
sg 15 
where the numbers and the letters have the same meaning as above. 
il 
An n-dimensional polytope of the degree of regularity = according 
B n 
to the scale given in my dissertation will be qualified, for 1 <p <n—1, 
Did 
1 
by the degree of regularity 5 according to the new scale, whilst 
((——— 
this degree would acquire the same value for both scales in the 
cases p=—0O and p=n, ie. for entirely irregular and for regular 
polytopes. For in the cases 1<p<mn—1 a polytope loses in the 
first of the two possibilities indicated by either and or a half characte- 
ristic, whilst the total number of available characteristics diminishes 
by a half at either side which changes the denominator n into n—1. 
In this paper I wish to take position with respect to the modifi- 
cation of my scale due to Dr. Scroure. Thereby I will have occasion 
to point out three different moments. 
1. Besides for entirely irregular and for regular polytopes the 
two scales coincide with respect to semiregular polytopes proper. 
For the supposition 
gives 
2p(n—1) = n(2p—1), 
Le. p = 3n and therefore 
n i 
So, if we arrange the polytopes of space S, in three groups, for 
which the degree of regularity is successively smaller than a half, 
equal to a half and larger than a half the modification proposed 
brings no alteration in these groups. Otherwise: in passing to the 
new scale the polytopes with a degree of regularity equal to a half 
14 
Proceedings Royal Acad. Amsterdam, Vol. XY. 
