MAXIMUM AND MINIMUM VALUES OF A LINEAR FUNCTION, ETC. 



57 



Hence the coefficient of p in (31) is always positive. It follows that the 

 values of .s throughout the neighborhood of A are greater than the value 

 at A which is a minimum. In like manner when Q is any point on the 

 hyperbolic arc D F oi (39) s has a minimum at B^ and when on the 

 hyperbolic arc D E ol (29) at C. 



In particular if § be at D then « = 0, ;8 = ^, 7 = | ; A' = 0, 

 B' = C = ^ir. Equation (33) gives s = ^ a, therefore r^ -\- r.^^ a and 

 P must be on side a. In fact it is the projection of A on a, for (35) gives 



v., = /( cot B, Tg = h cot Cj 

 where /( is the altitude from A. This is not a true minimum for s, 

 because s is constant and equal to -J a for all positions of P on side a ; it 

 is clearly less than the value for any point P not on side a. Similar re- 

 sults follow when Q is at E or F. 



