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UNIVERSITY OF VIRGINIA PUBLICATIONS 



III. 



19. Let A B C D be a tetrahedron, a, h, c the sides oi A B C and 

 f, g, h the edges from D to A, D, C. We seek the maximum or niinimum 

 value of 



s = ar, + ;8r, + yr3 + 8r,, (52) 



the i-'& being the radial coordinates of a point P to the corners A, B, G, D 

 and X, y, z, w its tetrahedral coordinates, a, ^, y, 8 are taken as the 

 tetrahedral coordinates of an arbitrarily fixed point Q. 



The identical relation existing between the content coordinates and the 

 radial coordinates of the same point is 



= a; r^^ + 2/ ''2^ + ^ ''3^ + '^ '''4,' 

 -\- xtj c- -\-xzb^ -\- y za^ -\-x wf^ -j-i/ ir g- -\- z w h-. (53) 



The second line of this equation we represent by a = 5 xyc-. remem- 

 bering that — (T is the power of the point with respect to the circumsphere. 

 The radial identity asserts that the segments .ri\, yi\, zi\, ivr^ laid off at 

 P in direction toward or from the corner A, B, C, D respectively according 

 as the coordinate x, y, z, w is positive or negative, are in equilibrium at P. 

 Otherwise directly from the definition of these coordinates it follows that 

 each of these segments is proportional to the sine of the solid angle between 

 the other three, which establishes the relation in question. We shall repre- 

 sent the radial identity 



= 0, 



by 



(54) 



^{ah cf glir-^r^r^r^)=:Q. (55) 



20. Lemma. — A sphere circumscribes a tetrahedron ABCD, at a point 

 P on the sphere in the trihedral angle D-ABC four segments a, p, y, 8 are 

 in equilibrium, directed to A, B, and from D respectively. Then 



s = ai\ + /3i\ + y>\ — Sr^ = 0, (56) 



since the ratios of i\, 1\, r^, i\ to the diameter through P are respectively 

 the cosines of the angles which the segments make with the diameter, and 

 the sum of the projections on the diameter vanishes. Also 



'S.xyzS- 



xyzw. 



(57) 



