MAXIMUM AND MINIMUM VALUES OF A LINEAR FUNCTION, ETC. 63 



dS 

 be zero at B when Q is on —;:— = the tangent to S = at B. It will be 

 dy 



zero at botli A and B when Q is at the intersection of these two lines, or 

 at the excenter C. The lines 



dS _ 8S ^ dS ^^ 

 8x d y D z ' 



the tangents to S = at the vertices are the sides of the triangle of the 

 excenters. It is not possible for s to be zero at A, B and C for any 

 position of Q. At no point on any of these lines is there an ordinary 

 maximum. At their intersections only can a, jS, y be in equilibrium satisfy- 

 ing the ordinary necessary conditions. 



15. The equation (14) which gives the value of s when an ordinary 

 maximum or minimum at x, y, z shows that x, y, z must make the expres- 

 sion on the right positive. That is P must be inside ^4 B C or a vertical 

 angle of ABC or one of the lunules between the circumcircle ABC 

 and the ellipse 2 x y c'- = 0. 



17. The conditions expressing the relations between P and Q are 

 interesting from the point of view of transformations. Thus, using (17) 



„ d cr ^ 8 a „ d (J 



ox y 8 y 



and putting <ti, o-j, u,, for the derivatives of o- with respect to x, y, z, we 

 have the transformations 



n. R V R tT rr 



(37) 



Limiting the radicals to positive sign, this transforms Q into P. Also 

 using (19), equations (30) transform P into Q. Therefore if the equatio?a 

 to the path of either be given that of the other can be written at once. 

 Eelations (37) transform the hyperbolic surface, bounded by hyperbolas 

 DBF, inside ABC into the triangle surface ABC. It transforms the 

 hyperbolic lunule between D and A^ into the circular segment in angle A 

 with chord B C. It transforms the hyperbolic triangle and the three 

 lunules into the surface and boundary of the circumcircle. It transforms 



