MAXIMUM AND MINIMUM VALUES OF A LINEAR FUNCTION, ETC. 



81 



When F =: or the point is on the circumsphere, the determinant in the 

 numerator equated to zero is the 'radial equation of the eircumspheric in 

 terms of the edges of the simplex. 



We can determine the power P in another way. Let (91) in which 

 we change n into n -\- 1 give V'-R'^ for a simplex in {n -\- 1) -space which 

 has the simplex in (91) for base and has a vertex T having radial coordi- 

 nates Pi, 0+2 with respect to the points of the base. Let r^ be the coordi- 

 nates with respect to the base of F the foot of the altitude (equal to h) 

 from T to the base. Produce the altitude to cut the (n -|- 1) -space spheric 

 again at a distance 8 from the base. Then the power of F with respect to 

 the (n -j- 1) -spheric is equal to the power of F with respect to the eircum- 

 spheric of the base and P = hS. Also 



V' = ]iV/ (n+l). 



. P^ _ (w+1)^ / 8 



• • V'^R'- ~ V^ yR' 



Let F be fixed and move T along the altitude to coincide with F, then 

 V = 0, R' = 00 and pi, n+i become r,, the limit of 8^R' is 2. 



(_l)n+l 



^(n\yv^ 



(93) 



Eepresent the determinant in (93) by A and that in (93) by D. 

 Making use of the value of V- in (90) there results the relation 



4A = I^=<^,,„ (94) 



a remarkable form of the radial identity. 



The results of this article furnish the means for attacking the general 

 problem; they have been deduced here because they are not of handy refer- 

 ence, and the writer would not know where to refer to the power equations 

 (92) and (93). 



VIII. 



26. In seeking the minimum (maximum) of any linear function of the 

 radial coordiuates rj of a point P with respect to the n -\- 1 corners of a 

 simplex of n dimensions, the function for examination can obviously be re- 

 duced to the study of 



s = 2air,, (i= 1, 2, . .., re+ 1) (95) 



