( 425 ) 



constituents (P),,^ , (P)„„ , (P)„3 which may be represented by the 

 symbol (P„, ; P,,.^ ; I\). In the same way we find by combining- 

 (P)„, and {I\ ■ 1\ ; P,.,) a pi-ismotope (P,,, ; 1\ ; 1\ ■ I\) witii four 

 constituents and finally, after having used (P)„ , a prismotope 



tuents. 



[P,,^ ; 1\ ; . . . \ P„) with p consti 



It is not difficult to show that the result is independent of the 

 order of succession in which the constituents are introduced in the 

 process mentioned. To that end we have only to demonstrate that 

 the interchange of (7^)„, and (P)^^ in the generation of (P,,^ ; P„J 

 does not influence the result. Let P be an arbitrarily chosen 

 point of the position {P)',,^-oï (P)„, in which the point of {P)„^ 

 coincides with the arbitrarily chosen point 0^ of (P),,^ , and let 

 O^ be determined by the vector equation 0^P= 00,, i.e. let 0^ 

 be (he point of (P)„„ corresponding to the point P of {P)'„^. Then 

 OO^PO^ is a parallelogram; so P may be considered quite as well 

 as the point of a new position {P)',,^ of (P),,^ corresponding to the 

 point 0, of (P)„j , this new position (P)',,^ of (P)„^ being obtained 

 by moving (P),,^ equipollent to itself in such a way that the point 

 of (P)„, coincides with the point 0, of (P)-„. Or, in connection 

 with the remark that 0^ and 0., are arbitrary points of {P),.^ and 

 (P),,2 , whilst OP is the resultant of the vectors 00, and 00, : "if 

 0, and 0, are arbitrary points of (P)„, and (P),,^ , the end point P 

 of the resultant OP of 00, and 00, is an arbitrary point of the 

 prismotope with the two constituents (P)„j and [P),,,". This can be 

 extended immediately to: "if 0,, 0, , . . . , 0,, are arbitrary points of 

 {P)n, , {P)v,, ■ • •, (P)«,, the end point P of the resultant OP of 



00, , 00, , . . . , OOp is an arbitrary point of the prismotope 

 (PR- • P V' 



This mode of generation shows clearly the irrelevancy of the order 



of succession of all the constituents. 



p 

 The prismotope (J\ ; 7\^ ; . . . ; 7^„ ) is a polvtope with ^m dimen- 



sion«; the space with this number of dimensions containing it is 

 completely determined by the spaces 6",,, , S„, , . . . , Sn with the com- 

 mon point 0. 



The aim of this paper is to determine the characteristic numbers 

 of the prismotope (P„^ ; P„, ; . . . ; P„ ). 



2. j^otation. We indicate the numbers of the vortices, edges, 

 faces, . . . , limiting poly topes (/)„-i of a poly tope P„ by the same 

 letter, say a, with the subsci'ipts 0, 1, 2, .... , n — 1 and represent 



