DERIVED FKOM THE EEGULAU POLYTOPES. 21 



original subject of e 2 (the square) is transformed by e x into an 

 octagon (fig. 13 b ) and now the octagon is moved out, in the case 

 e i e 2 C the linear subject of e x (the edge) is transformed by e 2 into 

 a square (fig. 13 c ) and now this square is moved out; in both 

 cases the result (fig. 13 d ) is the same, a WO. In general, for h > /, 

 in the case e k e l M^ 2) the subject M,} 2 ' of e k is transformed by e t 

 into an M k {2 '\ while in the case e ( e k M^ 2) the subject J// 2) of e l 

 is transformed by e k into an n — 1 -dimensional limit g { of the 

 import /. Here also the geometrical condition "that the two new 

 positions of any vertex shall be separated by the length of an edge" 

 makes the distance over which the second motion of any of these 

 pairs has to take place equal to the distance described in the first 

 motion of the other pair; i. e. if J// 2) is a limit of the limit M k (2) 

 of M n C2) and A is a vertex of that M t (2) , the segments described 

 by A in transforming M n (2) into the two polytopes e k e l M^ 2) and 

 e l e k M^ 1) are the two pairs of sides, with the length \/2(n — k) and 

 \/2(n — /), of a rectangle leading from A to the opposite vertex A'. 

 So we find the coordinates of A' by adding to the coordinates of A the 

 variations corresponding to the motions due to each of the opera- 

 tions e k , e t taken separately. So, in the case of three or more ex- 

 pansions we will have to use the extension of this rule to parallel- 

 opipeda and parallelotopes; to this geometrical composition of motions 

 always corresponds the arithmetical addition of influences. So the 

 general rule is proved. 



By the way we still find the theorem: 



Theorem XXXV. "The operation e k can still be applied to any 

 expansion form deduced from Mjjp in the symbol of coordinates 

 of which the n — k th and the n — h -\- I st digit, i. e. the k th and 

 the k -j-- 1 st digit counted from the end, are equal" 



This theorem enables us to find immediately the expansion symbols 

 of an expansion form deduced from M^ 2) with given coordinate symbol. 

 We show this by the example [5 4 4 3 3 2 2' 2 1 1] of art. 55. 



In [5' 4' 4' 3' 3' 2' 2' 2' V 1] five intervals occur, viz, if we represent 

 the p th digit from the end by d p between (d { ,d 2 ), (d,,d d ), (d 5 ,d 6 ), 

 (d 1} d s ), (d 9 , d i0 ). So we find e i e 2 e b e 1 e 9 M i0 . 



60. By means of the operations e k we can deduce from M t \ 2) all 

 the possible polytopes the square bracketed symbol of coordinates 

 of which winds up in a unit. If we wish to deduce from M^ also 

 all the forms with a square bracketed symbol ending in zero — which 

 is a desideratum as to the treatment of the nets — we have to 

 introduce the operation c of contraction. The subject of this contraction 



