MAXIMUM AXU MINIMUM VALUES 01' A LINEAU FUNCTION, ETC. 83 



sliows that the segments Xi r^ are equilibrated at P in the same way as are 

 the secrments "i. Therefore the relations 



x.,r. 



and each of these ratios by compounding them is equal to 

 '%x^Xip^-i _ s 



(lUl) 



(102) 



Also 



S ia,-/.Ti 



.•. S- = ^-^. %X,Xip^i, (103) 



-^^ = ^^= ... =-^, (1041 



PiOi p„a2 S 



.-. S- = a,-p,+ ... -^a\,,p^,,. (105) 



If Q be chosen a point of finite position such that each a is different 

 from zero, (101), (102), (103) show that it is impossible for s to vanish 

 unless "XxiXipi^i = and also Sai^/a^i is zero. That is unless P is a point 

 on the circumspheric where the second surface cuts it. We write these 

 surfaces o- = and o-' = 0. The surface cr' = contains the edges and 

 vertices of the simplex. It is a continuous algebraic surface of degree n. 

 We assume that if it cuts the spheric in a point P not a vertex that it will 

 cut the spheric in lines leading to the vertices along the surface of the 

 spheric. 



87. If n segments ai meet at P and have the resultant an+i and a 

 spheric be drawn through P cutting the n-\- 1 segments again at distances 

 rj, . . ., Tn and the resultant at p from P then 



2ai?"i = an+ip . 



The proof follows on dividing both sides by the diameter of the spheric, 

 as in three dimensions. 



Let L represent the (n-+l)th vertex of the simplex. Let % be 

 equilibrated at P. Through P and the other n vertices pass a spheric. 

 Produce LP to cut this spheric again in L, and let PL^ = p. Then 



2air, = a„,,p . (106) 



Represent the n common vertices of these two spherics by F^. . ., V^ and 

 their distances from L^ by pi, . . ., pn- 



