On the Surfaces of the Second Order. 313 



between the two systems of surfaces are therefore perfectly reci- 

 procal. From the equations (14) and (15) it is manifest that 

 the asymptotic cones of the hyperboloids of one system pass 

 through the focals of the other. 



16. The point S being the intersection of a given system 

 of confocal paraboloids whose equations are 



+ = x + h, , + = % + n, 



p q p' q 



(17) 



where p -/= q - q'= 4 (A - A')> and p -p" = q-q"=(h- A") : 

 suppose that another paraboloid A confocal with these, and 

 expressed by the equation 



/ 2 s 2 



y - + - = x + h , (18) 



Po o 



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 co-ordinates 

 , TJ, respectively, the equation ef the circumscribing cone will 

 be 



P ~Po P ~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 surface A, 

 provided it be confocal with the given surfaces. If the hyper- 

 bolic paraboloid be the second surface of the given system, the 

 parameter p' will be intermediate in value between p and p", 

 and the equation of the focal lines of the cone will be 



- + -77, = 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 



