76 



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G be any n -\- 2 points. In ( 72 ) move the point in succession to the 

 corners of the simplex, p becomes p^ in order and pr becomes pi, j, while 

 Sfcifcjpi", j remains unchanged. Thus when coincides with the /7th 

 vertex, 



Pp" ^ ^'iPi'p + • ■ • + ^'p-iPp"-1) p + '''p+iPp'+u p + • • • + ^ViP'n+ij p 2, 



thus n -\- 1 equations and in addition SA'r = 1. Eliminate the I's and 2, 

 there results the identical relation 



111 



1 ^ pi^o pi' 



1 Pl^2 p,- 



1 />1^3 P2' 







= 0, 



(76) 



1 p^'- p.- p.- ... 



there being n -)- 3 rows. This is a relation between the lengths of the 

 edges of the simplex and the radial coordinates of any point in space. We 

 refer to this as the radial identity, writing it briefly 



<^(pi,j,p,)=0. (77) 



To find the radius R of the spheric circumscribing the simplex, it is 

 only necessary in (75) to move the point pr to each vertex in succession, 

 and, eliminate K^ from the resulting equations. Hence 



R' = — i 



(78) 



If Pj pr, (r = 1, 2, . . ., /I + 1) are arbitrary numbers 



p = 2tr/Jr — SA-itjpi^i, (79) 



is the equation of a spheric in fc-coordinates. The coefficients p^ are the 

 powers respectively of the vertices of the simplex and p the power of the 

 point kr with respect to the spheric 



= MrPr — MiJCipi,^i. (80) 



This last is the equation of the orthotomic spheric of the n -\- 1 spherics 

 whose centers are the vertices of the simplex and the squares of whose radii 

 are respectively pr- The proof of this theorem is as follows. The condi- 

 tions that (70) shall be a spheric (72) are 



= 1. 



(81) 

 (82) 



