503 



y , ^ , / v 



P <1 P 1 „., 



2 2 (^') 



where p — p' — q — q' = 4{h— h'), and p — p" — q— q" 

 z=. A{h — h") ; suppose that another paraboloid A confocal 

 with these, and expressed by the equation 



^+-=^ + /'o, (18) 



Po qa 



is circumscribed by a cone having its vertex at S. Then if 

 the normals applied at S to the given system of surfaces, 

 taken in the order of their equations, be the axes of the coor- 

 dinates ^, r/, ^ respectively, the equation of the circumscribing 

 cone will be 



9"> 2 fi 



P—PO 13— PO P —PO 



showing that those normals are the axes of the cone, and 

 that the focal lines of the cone are independent of the sur- 

 face A, provided it be confocal with the given surfaces. If 

 the hyperbolic paraboloid be the second surface of the given 

 system, the parameter p' will be intermediate in value be- 

 tween p and p", and the equation of the focal lines of the 

 cone will be 



-7-^ + -^^, = 0, (20) 



p — p p — p 



which is the equation of a pair of right lines parallel to the 

 asymptotes of a section made in the hyperbolic paraboloid by 

 a plane parallel to the plane |^, since the quantities p' — p 

 and p' — p" are proportional to the squares of the semiaxes 

 of the section which are parallel to $ and Z, respectively. The 

 focal lines are therefore the generatrices of the hyperbolic 

 paraboloid at the point S. 



VOL. II. 2 u 



