DERIVED FROM THE REGULAR POLYTOPES. 31 



square is moved out; in both cases the result (fig. 4 d ) is the 

 same, tO. In general, for /• > /, in the case e k e t /S >(1) (n --j- 1) the 

 subject S {X) {Jc -f- 1) of e k is transformed by e t into an e t /S (i) (/c ~\- 1) 

 and now this e t S^(&-\~ 1) is moved away from the centre, while 

 in the case e t e k S^(n -j- 1) the subject # (1) (/-j- 1) of e, is trans- 

 formed by e k into an n — 1 -dimensional pólytope of the import / 

 corresponding to S (i) (I -\- \) which polytope is moved away from 

 O as a whole. Now it is evident that the geometrical condition 

 "that the two new positions of a vertex shall be separated by the 

 length of an edge" makes the distance over which the second 

 motion of any of these two pairs has to take place equal to the 

 distance described in the first motion of the other pair; i. e. if 

 /^(/-j- 1) is a limiting element of /S (i) (k ~\- 1) and A is a vertex 

 of that /S 7(1) (/ -\- I), the segments described by A in transforming 

 8 (i) (n-\- 1) into the two polytopes e k e, S(n -j- 1 ) and e l e k S(n -f- 1 ) 

 are the two pairs of sides of a parallelogram leading from A to 

 the opposite vertex A'. In other words: we find the true coordi- 

 nates of A' by adding to the coordinates of A the variations cor- 

 responding to the motions due to each of the operations e k and e x 

 taken separately. 



Taking for tf (1 >(v£-f- 1) the simplex A ± A 2 . . . A k + i , for £ (1) (/-f 1) 

 the simplex A i A 2 . . . A l + 1 and for A the point A i we have to 



n 



vary the coordinates 1, 0, 0, ... of A so as to admit two more 

 unit intervals, one between the k -j- I st and the h -f- 2 nd , an other 

 between the / -f- I s ' and the l~\- 2 nd digit. If then afterwards we 



k—l n — k 



pass to the zero symbol we get (3 22 . . 2 1 1 . . 1 00 .'. 0), what 

 was to be proved. 



Now we have still to add that the proof for the composition of 

 three and more operations of expansion runs entirely on the same 

 lines. In the case of three operations we will have to compose 

 three displacements according to the rule of the diagonal of the 

 parallelopipedon , in the case of more we will have to use the 

 extension of this rule to parallelotopes. To this geometrical compo- 

 sition of motions always corresponds the arithmetical addition of 

 the symbol influences, where the order of succession is irrelevant; 

 this arithmetical addition leads to the creation of new unit inter- 

 vals independently. So the general rule is proved. 



The preceding developments lead to a new theorem , viz : 

 Theorem IX. "The operation e k can still be applied to any poly- 

 tope deduced from the simplex in the zero symbol of which the 

 k -f- l sf and the k -f 2 m ' digit are equal." 



