62 UNIVERSITY OF VIRGINIA PUBLICATIONS 



easily seen by resolving along r^. The triangle A'B'C is similar to A B 0. 

 The function s is positive at all other points in the plane and is therefore 

 a pseudo minimum along B G. Similarly when Q is at excenter B^, (7, 

 then s is zero and a pseudo minimum along arc A C, A B respectively. 



(2) . When Q is on hyperbola E D A^ in angle A then P is at vertex G. 

 The discussion is the same as if Q were on the arc F E B^^ but in angle B, 

 then ^ is at 4 and the conditions can be examined by making ^ negative 

 in (30). The same reasoning as in § 10 shows the coefficient of p positive, 

 and therefore s at 4 is a maximum or minimum according as its value 

 — ¥c + c'b is positive or negative, that is according as Q is between E 

 and Bi or B^ is between E and Q. 



Therefore when Q is on hyperbola E D A^ between E and A^ s has a 

 singular maximum at 0, when Q is on the prolongation of the are E A^ a 

 singular minimum at C. 



(3). If Q is on the branch of the hyperbola F E A.^ in angle A then P 

 is at A and as is obvious from (30) s has a singular maximum at A. 



(4). In the areas unconsidered in the vertical angle D the conditions 

 for positive i-'s fail to give results; these are provided for by the cases 

 already considered by corresponding changes in signs of a, P, y. In case 

 Q is in angle C outside vertical angle D then one of a', b', c' is greater than 

 the sum of the other two and no ordinary maximum or minimum exists; 

 there will be a singular maximum or minimum at a vertex such that the 

 straight line 



X i\ -\- y r„ -\- z r^ ^ — (>■' i\ + V 1\ -\- c' r, 



meets a hyperbola in x', y', z' corresponding to which point s has a singu- 

 larity at that vertex. If we investigate the conditions that the singular 

 maximum of s occurring at the vertices shall be zero, .9 will be zero at 

 A, B, C respectively when 



C l3 + by = 0, 

 f a -|- a y = 0, 



b a + a 13 = 0. 

 Eepresenting the circumconic 



xyc-\-xzb-\-yzc^O 



dS 

 hj S = 0, then s = at 4 when Q is on -^- = the tangent to S — 



at A. This straight line passes through the B and C excenters. Also s will 



