MAXIMUM AND MINIMUM VALUES OF A LINEAR FUNCTION, ETC. 77 



This is tlie radial equation of a linear locus perpendicular to the edge 



Prj r+i- The sum of all such equations vanishes, identically showing that 



these linear loci meet in a point, the center. In like manner (80) and 

 spheric 



coincide provided 



Pl!zi:?l= . . . = p!£±iZZ^ = 1. (83) 



Pi Pn.i 



Whence as before 



P't P"r+1 = Pr Pr+i- 



The assigned numbers pr assign the same center as before, independent 

 of p. The equations (81) and (83) show that 



P't — p^ = Pt — p and p't — R^ ^ Pr, 

 hence 



p = p'--R^ 



is the power of Av with respect to (80). The minimum value of pis when 

 p = then p = — R-. Also 



pr = pV — R^, 



shows that pr is the power of Fr with respect to (80). Equation (79) is 

 then a spheric concentric with (80), the power of an}^ point on which with 

 respect to (80) is p. 



To find the content 7 of a simplex, we first show^ that if V is the 

 content of the simplex in (n-1) -space determined by the vertices but one 

 of the n.-space simplex, then V = pVy'n. Let p be the perpendicular 

 distance from the (w + l)th vertex to the linear locus through the other 

 vertices. Let z be the perpendicular on a parallel linear locus, and v the 

 content of the (w-1) -space simplex determined by the points where the 

 edges from the (w+l)th vertex meet the parallel locus. Then, be- 

 cause of similar figures 



+ ■ 



V ' p°— 1 

 P ^ P 



f ^ dz=-^ f ^n-1 dz = ^ pV. (84) 



'See a footnote, p. 326, in Mr. Sharp's paper. 



