60 UNIVERSITY OF VIRGINIA PUBLICATIONS 



first. Hence our convention of keeping the r's positive and cliangiug the 

 signs of a, /?, 7, and in fact only one of them, separates the sheets of 

 this surface of degree 8, and will serve to find the maximum or minimum 

 ordinate of each. The complete surface has no maximum or minimum at 

 A, B, C when that point is singular on a sheet, but passes through that 

 point tangent to a cone of second degree there. The convention as to the 

 signs of the ^•'s being positive cuts off one nappe of this cone, giving the 

 singular maximum or minimum of the incomplete surface we have found. 



13. We consider now the remaining case when one of a, p, y is nega- 

 tive, sa}^ a, or (? is in the vertical angle D of the triangle D E F. 



In the formula for the position of P at which an ordinar}' maximum 

 or minimum may occur, the negative signs are to be taken throughout. 

 Since the r's are positive the point P must occur outside ABC and in the 

 angle C. Eeferring to § 6 for the relations between the angles A', B', C" 

 and the angles subtended by the sides of A B C at P, we recognize that the 

 branches of the three hyperbolas (29) which lie in the vertical angle D 

 are such that the first, whose equation is now 



y- -\- z- — x^ — • 2xy cos A = 0, 



contains all points Q for which .i'=.l; the second and tliird. whose 

 equations are the same as before, contain respectively all points for which 

 -B' = 5 and C = C. It is easy to see, on trial, that these three hyperbolas 

 pass through the point — a:b:c which is the A excenter of A B C, which 

 point we represent by ^1^. 



(1). If (> is in the closed lenticular space between the two hyperbolas 

 FDA, and FDA, then 



A'<A, B'>B, C">0, 



Pi, X. a are negative, <x, s are positive, the ?-'s are positive. The discrimi- 

 nant (38) is negative, and hence s has an ordinary maximum at a deter- 

 mined point P in the segment of the circumcircle of ABC having B C 

 for its chord. 



(2). Passing Q through A, into the open space between the same 

 hyperbolic arcs,. 



A'>A, B'<B, C'<C, 



p, is -j-, y, and p^ are — , s is negative, r is — , y and z +, and cr is 

 negative. The discriminant (29) is positive. Hence s is a minimum at a 

 point P in the angle A outside the circumcircle A B C. 



