MAXIMUM AND MINIMUM VALUES OF A LINEAR FUNCTION, ETC. 



79 



Observe the values of fcr in (71); each is equal to the content of a 

 simplex whose vertex is .r, //, x, . . . and whose base is a simplex face of 

 the n-space simplex, divided by the content of the n-spaee simplex of 

 reference. Hence the I'a are the content coordinates of a point with refer- 

 ence to the simplex of reference. 



We shall now abandon cartesian coordinates and in the future use 

 radial and content coordinates as being better suited to the problem before 

 us. We shall represent the coordinates of a point P referred to the simplex 

 of reference by ri for radial coordinates and by Xj for content coordinates, 

 r= 1,2, ...,n + l. 



The equation of a sphere in radial coordinates r^, i\, . . ., whose radius 

 is R and whose center is given by the content coordinates a:,, x^, ... is 



= Sa^iJ'i- — %XiXipi-j — R-. 



The square of the distance of any point Vi from x^ is 

 8- = S^iri- — 'SiXiXipi^i. 



(86) 



(87) 



■ R- of the point 



with respect to the sphere (86) is 

 R-. 



The power 8 

 therefore 



p = •S.Xii-r — ^XiXjpi'i — R-. (88) 



We shall now find the content of a simplex in terms of its edges. Let 

 p be the length of the perpendicular from the (n -\- l)tli vertex on the 

 base (the altitude of the simplex), and let Xi, r^ (i = 1, 2, . . ., n) be the 

 content and radial coordinates of the foot of the perpendicular with re- 

 spect to the base simplex. Let 



<T = 2.x'ia;]pi"i, 

 then 



pr,n.. = n- + r, 'v=-^^-- 



Eliminate Xi and o- from the n -\- 2 equations 

 %Xi — l = 0, 



(89) 



-i 



