205 
the same length *) admits one characteristic of regularity more than 
its vertex polytope, i.e. if the latter is n-dimensional and admits the 
degree of regularity sie the former must admit the degree of regularity 
n 
pd 
n+1 
does not bring any alteration. If we build up an »-+1-dimensional 
polytope by starting from a given n-dimensional vertex polytope, 
the n+1-dimensional polytope will possess all the characteristics of 
regularity of the n-dimensional one, each of these adapted to limiting 
elements of one dimension higher, and moreover it obtains at the 
beginning of the series two new halves of characteristics, i.e. equal 
vertices and edges of the same length. Finally the denominator like- 
wise increases by unity, the new polytope admitting one dimension 
more than its vertex polytope. 
In my dissertation I had to point out an exception to this rule, 
presenting itself in the case p=0O, i.e. when the vertex polytape 
In this rule the indicated modification of the scale evidently 
instead of 
0 
is irregular. For in that case — passes into 
n 
n n-+-1 
So the vertex polytope of the semiregular polyhedra of the table 
— ie. “the vertex polygon” here — is an isosceles triangle for the 
numbers 1-—5 and 14, an isosceles trapezium for 8, 9, 15, ascalene 
triangle for 10, 11, a symmetric pentagon for 12, 13 and therefore 
0 re 
the degree of regularity a of the vertex polygon has to lead to = == 4 
in the cases enumerated. This exception now disappears by intro- 
duction of the new scale of Dr. Scnoure; for according to this scale 
0 ; i. 
= passes into 5 in these cases. 
On account of the latter important advantage of the new scale 
over the old one I wish to accept the first. Therefore I insert 
finally a second table in which the polydimensional polytopes with 
a degree of regularity equal to or surpassing } are enumerated with 
addition of their degree of regularity according to the new scale. 
The superscripts S, represent the number of the n-dimensional 
limits of the polytope. The character of these limits is indicated by 
notations, the meaning of which is partially clear by itself or by 
the first table of this paper. Moreover we may state the meaning 
of the following symbols: 
1) The latter has been supposed tacitly on p. 129. 
MA 14* 
