MAXIMUM AND MINIMUM VALUES OF A LINEAR FUNCTION, ETC. 75 



Let pr be the distances from the arbitrarily chosen origin to the corners 

 of the simplex, and let pi, j,(i, / = 1, 2, . . ., w -f 1) be the lengths of the 

 edges of the simplex joining the corners two and two, the number of edges 

 being ^{n -\- 1) {n -\- 2) . Squaring the coordinates x, y, z, . . . 



X- = %h-x,- + 2-S.l-il-yXiXi. 



Eeplace l\- bv its equal value 



hr — hh, — ... — hh_, — /,v/>v,i — ... — IcrK,^. 

 Whence 



X- = %l\xr — SA^itj (Xi — Xj ) -, 



with precisel}- similar values for y-, z-, . . . 

 On adding 



p"- = :^hp,"- — M,hp,%i, (72) 



where p = OG, expressing the distance between two points 0, G, one given 

 by the radial coordinates, the other by the ratio coordinates with respect 

 to the simplex.^ The orthogonal coordinates having now disappeared, 

 (72) for p and pr constant is the equation in fc-eo6rdinates of a spheric 

 whose center is given by the radial coordinates p^ and whose radius is p. 

 Also for hr and p constant (73) is the radial equation of a spheric with 

 center l\ and radius p. If p = 0, then 



= 2fcrpr— SA-ifcjpi^j, (73) 



is an identical relation existing between the "content" (A',.) and radial 

 (pr) coordinates of any point in space. 



If coincides with the center of the circumspheric of the simplex, 

 Pr = B the radius of the circumspheric. Hence the equation in l\ coordi- 

 nates of the n-dimensional spheric circumscribing the simplex is 



= — 2fcifcjpi^^ (74) 



If Kr are the coordinates of the center, the equation to this spheric in the 

 radial coordinates pr is 



Q = -S,Erpr- — 2'R-. (75) 



There is an identical relation existing between the mutual distances con- 

 necting any » -|- 2 points in n-space. Let the corners of the simplex and 



'This is a generalization of Stewart's theorem in elementary geometry (1763). 

 Two points A, B on a straight being given, the distance from any point to a 

 point G in the line (dividing A B in given ratio) is given by 



0A-. BG + 0B-. GA + 0G\ AB + AB .BG.GA = 0. 



