BY A SPACE EOTATING ABOUT A PLANE. 15 



tions of their vertices consist of A', A and eight points dividing 

 AA' into nine equal parts. So we easily convince ourselves of 

 the fact that the 64 vertices of the 27 cubes project themselves 

 into the indicated ten points l ) and that the projection of any of 

 the 27 cubes on A A' covers a third part of that line limited by two of 

 these ten points. Returning to the projection of the 27 eightcells of 



*) If the three edges of the block of cubes concurring in A are taken as positive 

 axes of coordinates and we suppose the edge of the cubes to be unity, the coordinates 

 of the vertices are (p, q, /•), where p, g, r are to be chosen from 0, 1, 2, 3. So, if 

 n (Pi 1-, »') represents the number n of permutations with definite j>, q, r and we join 

 together the numbers n(p, q, r) for which p + q + r has the same value, we find 

 groups of vertices lying in the same plane x + y + ~ = constant, i.e. points with a 

 common projection on AA' . In this manner we get 



1 (000), 3 (100), 3 (200) + 3 (110) , 3 (300) 4- 6 (210) + 1 (111), 

 6 (310) 4- 3 (220) + 3 (211), etc. 

 i.e. groups of 1, 3, 6, 10, 12, 12, 10, 6, 3, 1 vertices projecting themselves in the 

 ten different points on AA'. 



This result admits of the following general extension : 



"If the (p-\-l) d vertices of a block of pd measure-polytopes of d-dimensional space 

 (arranged in the form of a measure-polytope of p times the size) are projected on to a 

 central diagonal of the block, the projections are the endpoints of that diagonal and 

 the pd — 1 points that divide it into pd equal parts. Moreover the numbers of the 

 projections coinciding respectively with these i>d -f- 1 points are the coefficients of the 

 successive powers of x in the polynomial (1 -f- x -j- x % +. . . .-J- xP)''." 



No doubt this theorem can be proved in different ways, e.g. by the induction from 

 p — 1 to p and d — 1 to d, or more simply still by the generation of the configuration 

 of the vertices, proceeding from a row of p + 1 equidistant points to a row of p -f- 1 

 squares, etc. Of the demonstration connected with the manner in which the theorem 

 was found we can only give a summary sketch here. It consists of three parts. In the 



first part we consider the special case p infinite, in which the block fills up the 9 j th 



part of d-dimensional space corresponding to the positive sense of the d axes OX t , 

 0X 2 ,. . . -OX d \ in that case the result depends, in the way indicated in the theorem, 

 on the form (1 — x)—'K In the second part we show that the finite block of the theorem 

 may be considered as the algebraic sum of 2 ( ' infinite blocks equipollent to the infinite 

 block of the first part but showing with regard to the position of the vertex at finite 

 distance that originally occupied the origin the effect of a parallel translation, any of 

 the vertices of the finite block forming the origin of one of these 2'' infinite ones; in 

 this algebraic sum the sign of any of the 2' 1 infinite blocks is positive or negative 

 according to the number of coordinates of its origin differing from nought being even 

 or odd. In the third part we prove that this sum of infinite blocks can be represented 

 by (1 — i?) d , if Bk represents one of these blocks, the system of coordinates of which 

 origin consists of k units and d — A- noughts (1 standing for B°, i.e. for the infinite 

 block in its original position) and then it is clear that this geometric composition of 

 the 2' 1 infinite blocks to the finite one is to be translated algebraically by multiplying 



(1 T p+ \ \(l 

 — j— J or (l+x+x 2 4- . . .xp)'I. 



We may add that the general theorem given above is also an extension of the pro- 

 blem in how many ways the word abracadabra written in a triangle may be read, or 

 in how many ways the king of the chessboard can march from a given row to any 

 row k squares higher (see E. Lucas, "Théorie des nombres", p. 13, 14). 



