AKM.Y IK' '/ "'// /'//I- (<-H. XI. 



Since in equation [To] the product mm' is positive, 

 it follows that the angles which conjugate diameter- make 

 with the transverse axis are both acute, or Ix.tli obtuse. 

 lion 01 ar, 



if m < -, then ml > - ; 

 a a 



and the diameters lie on opposite sides of an asymptote. 

 ite diameters lie in the same quadrant formed /// 

 the axes of the hyperbola, on opposite sides of the asymptote 

 (cf. An. i:>5(a)). 



"(8) An asymptote passes through the center of an hyper- 

 bola, hence may be regarded as a diameter. Its slope is 



b b 



tfi = -, .-. w' = -; 

 a a 



hence, an asymptote regarded as a diameter is its own conju- 

 gate ; it may be called a self -conjugate diameter. 



This is a limiting case of (7) above. 



) It follows from this last fact that if a diameter inter- 

 sects a given hyperbola, then the conjugate diameter does 

 not intersect it, but cuts the conjugate hyperbola. It is 

 customary and useful to define as the extremities of the 

 conjugate diameter its points of intersection with the conju- 

 gate hyperbola. With this limitation, it follows from (a) 



his article, that, as in the ellipse, each of two conjv 

 diameters bisects the chords parallel t<> tin 1 , other. 



(?) A.s a limiting case of this last prop.it inn. also, ii [fl 

 evident that the tangent at the end of a diameter is parallel 

 to the conjugate diameter. 



By reasoning entirely analogous to that i^m-ii in An. L55, 

 for the ellipse, properties similar to those there given may 

 for tin- hyperbola. They are included in tin- 

 following exercises, to be worked out by tin- student. 



