448 



the grasp of geometry. Now as the different conic sections 

 may (with the exception of the circle) be described in piano 

 by the motion of a point whose distance from a given point 

 bears a constant ratio to its distance from a given right Hne,* 

 it is natural to suppose that there must be some analogous 

 method by which the surfaces of the second order may be 

 generated in space. Accordingly I have sought for such a 

 method, and I have found that (with certain analogous ex- 

 ceptions) every surface of the second order may be regarded 

 as the locus of a point whose distance from a given point 

 bears a constant ratio to its distance from a given right line, 

 provided the latter distance be measured parallel to a given 

 plane ; this plane being, in general, oblique to the right Hne. 

 The given point I call, from analogy, a focus, and the given 

 right line a directrix ; the given plane may be called a direc- 

 tive plane, and the constant ratio may be termed the modulus. 

 To find the equation of the surface so defined, let the 

 axis of IS be pai-allel to the directrix ; let the plane of xy 

 pass through the focus, and cut the directrix perpendicularly 

 in A, the coordinates being rectangular, and their origin ar- 

 bitrarily assumed in that plane; and let the axis of y be pa- 

 rallel to the intersection of the plane of a-y with the directive 

 plane, the angle between the two planes being denoted by (p. 

 Then if we put xi, y^ for the coordinates of the focus, and 

 x<i, y2 for those of the point A, while the coordinates of a point 

 S upon the surface are denoted by x, y, z, the distance of 

 this last point from the focus will be the square root of the 

 quantity 



(a:-x,f + (y-7j,f^z'; 



and if a plane drawn through S, parallel to the directive 

 plane, be conceived to cut the directrix in D, the distance SD 

 will be the square root of the quantity 



* This method of describing the conic sections is due to the Greek geometers. 

 It is given by Pappus at the end of the Seventh Boolt of his Mathematical Col- 

 lections. 



