60 



and a fixed straight line a. If A is to lie on the quadratic surface 

 vviiich a describes wiieii it revolves round a straight line /, the circle 

 which A describes when it revolves round /, must cut the straight 

 line a; accordingly / must lie in the plane which bisects perpendi- 

 cularly the straight line joining .1 to a point of a. These planes 

 touch a parabolical cylinder that has for directrix the parabola of 

 which A is the focus and a the director line and the generatrices 

 of which are peipendicular to the plane {A,a). 



If a revolves round a straight line crossing it at right angles, a 

 plane is produced which, completed by the pUme through A parallel 

 to it, gives another 0^ that satisfies the conditions mentioned, hence : 



if three of the four points Ai are collineav, F splits up into a 

 pencil of rays ivith the axis at infinity, and the tangents to a para- 

 bolical cylinder. 



