78 



UNIVERSITY OF VIRGINIA PUBLICATIONS 



In 1, 2 and 3-space the contents of the respective simplicissima are 

 known to be 



a^i, 2/i, ^1, 1 

 x„, y^, z„, 1 



•^35 2/35 ^35 1 

 ^■45 2/4; ^j; 1 



+ 1) be the orthocartesian coordinates of 

 the n points fixing the base of a simple^ in n-space whose vertex is 

 »!, 2/ij ^15 • • ■ ■ Assume the content F' of the base in its (w-l) -space to 

 be conformable with the expressions for the content in 1, 2, 3-space, and 

 that 



Let 2/r, 2r, . . . (»• = 2, 3, 



7' = 



(n-1) ! 



n+i; -"n+ij 



Then the content Y of the simplex in n-space is, by (84), 



T =-- 



2/n 



For A is absolutely invariant under any transformation by rotation and 

 translation of orthogonal axes. Choose the linear locus through the n 

 points of the base 



2/r, 2r, . . . (r=2, 3, . . ., Jl + 1) 



as the coordinate reference locus x = 0, and the perpendicular from x-^, y^, z^, 

 on it as the corresponding axis. Then x^ = p and 



2/1 ^ 2i ^ . . . = 0, x„ ^ x^ ^ . . . ^ a;n+i = 0. 



Hence A is identical with the determinant in question, and for any 

 integer n 



V = 



(85) 



