MAXIMUM AND MINIMUM VALUES OF A LINEAR FUNCTION. ETC. 61 



(3). Pass Q over the hyperbola E D A^ into the open space between 

 that hyperbola and the hyperbola F A^ E. Then 



.4'>.4, B'<B, C'>C, 



Pi, P3 are positive, p„ negative. The r's are unlike signed. There is 

 therefore no ordinary maximum or minimum value of s in this case. 



(4). In like manner for Q in the open space between hyperbolas 

 FDA^ anAFA^E 



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



Pi, P2 ^i"6 positive, Ps negative. The j-'s are unlike signed and there is no 

 ordinary maximum or minimum. 



(5). Q in the open space between the straight line ED and the 

 hyperbolas EI)A^,FA^E, 



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



Pi, p„ are positive, p^ negative, the r's are unlike signed. 



( 6 ) . § in the open space between the straight line F E, hyperbolas 

 FDA„FA^E, 



A'<A, B'<B, OG, 



and similar conditions hold. Hence in this and the preceding case there 

 exists no ordinary maximum or minimum of s. 



These cases conclude all cases of the investigation and determination 

 of the ordinary maximum or minimum values of s. For corresponding 

 treatment of the spaces in the vertical angles E and F result in similar 

 conclusions by interchange of letters. 



14. We consider now the singular cases of maximum or minimum 

 values of s. 



(1). When Q is on one of the hyperbolas (29), P is at a vertex of 

 A B C. The three hyperbolas pass through each of the excenters oi A B 0. 

 Wh.2n Q is at the A excenter A^ then A' = A, B' = B, C' = C. The 

 powers Pi, p„, ps are all zero, and x, y, z are , / , indeterminate ; s is 

 zero, o- is zero, and the locus of the indeterminate point P is the circum- 

 circle. In fact the locus is in areal coordinates 



— aj-i + &r2 + cr, = 0, (36) 



Ptolemy's equation of the arc B C of the circumcircle in angle C. At 

 each point of this arc the segments — a, &, c are in equilibrium, as is 



