p^- = 



24:T, 



MAXIMUM AND MINIMUM VALUES OF A LINEAR FUNCTION, ETC. 71 



Produce the lines through P and the vertices A, B, C, D to meet the 

 four spheres again in points A^, B^, Cj, D^ respectively. Then 

 p^ = i\. AA^, etc. Therefore by (62) 



s = a.AA^ = p.BB^ = y.CC^ = 8.DD,. (64) 



At P on the sphere ABCP the segments a, p, y, 8 are in equilibrium, 

 the first three directed toward A, B, C respectively, and 8 being directed 

 away from D^ when P is assumed to be inside ABCD and a, p, y, 8 positive 

 numbers. Hence by the lemma, § 20, 



ar, + pr^ + yr, — 8. PD, = 0. 

 Move P in succession to A, B, C along the spherical cubic. There resuH 



AD^^ f' + -y\ BZ)^=J^'l±I^, CZ?,=^4^- (65) 

 8 8 8 



The tetrahedron ABCD^ is now completely known. If T^ is its volume then 



288r/ = ${a, b, c, AD^, BD^, CD-,), (66) 



1111 



1 c= &= AD,- 

 1 c- a- BD,- (67) 

 1 b- a- CD,- 

 1 AD,^ BD,^ CD,^ 



Hence the powers p,, p^, p^, p^ are known, and the coordinates of P and s 

 known, by similar treatment of the points A,, B-,, C-,. While T^ and p^ 

 have been written at once in terms of known numbers, DD,, etc., might have 

 been determined as a root of the quadratic 



*(a, b, c, AD„ BD„ CD,, f, g, h, DD,)=0, 

 thence s by (6-1). The radius of this sphere can be found by a quadratic, 

 thence the distance of the center from D hj a. quadratic, and then p^ from 

 these values. 



22. As to the existence of a maximum or minimum value of s, if 

 a, p, y, 8, i. e. Q, is represented by any point inside the tetrahedron, then s 

 at P is uniform, finite and positive at all points in the finite region and 

 infinite and positive at infinity; hence there must exist a minimum, for Q 

 outside it is not so evident. 



The six partial derivatives of s of second order are easily found froir 

 (59). The two typical ones are 



the others follow on change of letters. 



