84 UNIVERSITY OF VIRGINIA PUBLICATIONS 



Move P from the critical point to the&e n points in succession, preserving 

 the equality (106) at all points of the path. Therefore n equations 



+ O.2P12 + "3P13 + ■ • • + O-nPm = Pittnti 



°-ip2i -(- + a3p,3 + . . . + anpan = P2"n+i I (107) 



"iPni + a,pn2 + "sPns + • • • + ^ poan+i 



where pi, j are edges of the base of the given simplex and are known, p, 

 are the unknown edges of the auxiliary simplex and are determined by 

 (107). Hence the volume, radius, power of a point with respect to this 

 auxiliary circumspheric, can be at once written by § 25. Hence the 

 power of L with respect to it is known, this is the power we called pn^^ 

 in (100), (104), (105). In like manner the powers p^, ...,pn of the 

 remaining vertices of the simplex with respect to their auxiliary spheres 

 are known, and hence s is known by (105), the r's are known by (104) 

 and the x's by (101). 



38. Since the segments "i must be equilibrated no one can be greater 

 than the sum of the others. The linear loci 



±x^±: x^ ±: ... ± Xn^^ = 0, (108) 



which pass through the mid-points of the n concurrent edges at a vertex 

 mark off an interior and exterior region in which the point Q must occur. 

 On the boundary of this region 2ai = and the corresponding point P 

 is at infinity. There can be written as in 3-space a hyper-hyperboloid a 

 sheet of which passes through the mid-points of concurrent edges such 

 that when ai is on the surface F is at a vertex and there s has a hyper- 

 teat minimum or maximum. 



39. By putting 8 = Saiajpij, the condition that s shall vanish at a 

 vertex is 



-|-^=0 (t=l,2, ...,w-(-l). (109) 



The discriminant of these n -|- 1 equations does not in general vanish. 

 Any n of tliese equations furnishes a point Q for which s = at the corre- 

 sponding n corners of the simplex. If s vanishes at the first n vertices 

 and the corresponding values of "i determined in terms of the edges from 

 (109) are 4i, then the equation 



S4,r, =4n^ir„,i (110) 



is the radial equation of a path on the spheric surface along which A\ are 

 equilibrated, corresponding to the ptolemaic equation of the arc of the 

 circle, and along which s has a pseudo-minimum value. 



Tiasi Lawn, March, 1909. 



