83 UNIVERSITY OF VIRGINIA PUBLICATIONS 



t^'here Sai = 1, and ai are the content coordinates of a point Q whose 

 position is determined by the arbitrar}'' constants of the linear function. 

 If s' be the value of s at a point P' distant p from P and pTi is the angle 

 which P'P makes with the line Joining P to the i th vertex of the simplex, 

 we can expand s' in terms of s and the powers of p by Taylor's formula 

 and write 



ai 

 s' = s + p2ai cos(p7-i)+|/D-2 sm'(pri)-]- R. (96) 



ri 



The conditions for a min-niaximum are first 



SaiCos(pri)= (97) 



for all directions of p, or that the sum of the projections of a; (laid off at 

 P along Ti) on an arbitrary line shall vanish. That is the segments thus 

 constructed must be equilibrated at the critical point of ordinary position. 

 Second 



2- 



-^ sin=(prO (98) 



must keep its sign unchanged for all directions of p. This is ^uite obvi- 

 ously the case when we convene the r's positive and ai a point inside the 

 simplex. A minimum does exist for the positive one valued function s 

 at some point ri in the finite space. Our object is to find that point which 

 fulfils the necessary condition (97) and when determined the character 

 of s there can be determined by (98). 



Assuming that a critical point P exists in tlie simplex, let its content 

 coordinates be Xi. 



The equations of the n -\- 1 spherics, each of which passes through P 

 and n at a time of the vertices of the simplex, are 



Q ^ XmPm 'SiXiXjpi-j (99) 



m ^ 1, 2, . . ., n -\- 1. The coefficient pm is the power of the mth vertex 

 of the simplex with respect to the sphere through P and the other vertices. 

 The radical axes of these spheres, two and two are 



.TiPl = X^p^ = . .. = .1V,Pn+i, (= 'S.XiXipi'j = 0-) (100) 



and they meet in P. 

 The identical relation 



= ^Xii-i- — <j 



