1 8 ANALYTICAL TREATMENT OF THE POLYTOPES KEGULAKLY 



all points of 8 ni that the expression Hœ t is equal to the sum of all 



the digits of (P) tl , form a system of n — d mutually independent 

 equations, representing therefore, in accordance with the result of 

 the first part of the proof, a space 8 d9 bearing the [P) d found above. 

 If we write this system of equations in the form: 



A-, A-, 4- A-., A-, + A-, + ... + kn-d 



I>; =p ± , Z 00; = p ± -f" > 2 , . . . , E œ { = p { -\~p 2 "f . . . -\rP n _ d , 



i = 1 i=l i = 1 



it is evident that each of the equations represents an n — 1 -dimen- 

 sional limiting space of (P) n , the constant of the right hand member 

 being a maximum. 



As we remarked already the crux of the proof of this part lies 

 in the true interpretation of the expression "adjacent digits". It 

 cannot be replaced by the condition that all the syllables should 

 be formed according to the first theorem. We show this by means 

 of two simple cases concerned with the determination of faces of 

 threedimensional polyhedra. In the case of the (2110)= CO the 

 hexagon (210) (1) is no face but a section, likewise in the case of 

 the (1100)= O the square (10) (10) is no face but a, section. In 

 both cases the syllables satisfy the conditions of the first theorem ; 

 but the impossibility of putting the syllables behind each other so 

 as to obtain the order of succession of the digits of the pattern 

 vertex implies the impossibility of finding an equation where the 

 constant that is equal to the sum of some of the coordinates is 

 either a maximum or a minimum. Under the five polyhedra T, O, 

 tl\ CO, tO which can be represented by a symbol with four digits 

 (compare the small table of art. 3 for n = 3) the and CO are 

 the only ones with sections p k and p 6 with sides equal to the edges; 

 at the same time they are the only ones with four edges through 

 each vertex. 



c). By means of the theorem we obtain all the limits {P) d of (P) n . 



It is always possible to represent any limit (P) d of (P) n by n — d 

 equations of spaces /S 7 n _ 1 containing n — 1 -dimensional limits (/) /( _i 

 of (P) n \ as the vertices of this {P) d are also vertices of (P) a , this 

 system of equations will be in accordance with the symbol of (P) n , 

 i. e. this system must be included into the set of systems of equa- 

 tions provided by the theorem. 



11. We apply the theorem III to an other fivedimensional form 

 (321100), showing at the same time how we can determine the 

 numbers of all the different limits. 



