262 On the Surfaces of the Second Order. 



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 line,* it is natural to suppose 

 that there must be some analogous method by which the sur- 

 faces of the second order may be generated in space. Accord- 

 ingly I have sought for such a method, and I have found that 

 (with certain analogous exceptions) 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 pa- 

 rallel to a given plane ; this plane being, in general, oblique to 

 the right line. The given point I call, from analogy, a focus, 

 and the given right line a directrix; the given plane may be 

 called a directive plane, and the constant ratio may be termed 

 the modulus. 



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

 of z be parallel to the directrix ; let the plane of xy pass through 

 the focus, and cut the directrix perpendicularly in A, the co- 

 ordinates being rectangular, and their origin arbitrarily assumed 

 in that plane ; and let the axis of y be parallel to the intersec- 

 tion of the plane xy with the directive plane, the angle between 

 the two planes being denoted by $. Then if we put x^ y^ for 

 the co-ordinates of the focus, and x 2 , y 2 for those of the point A, 

 while the co-ordinates of a point S upon the surface are denoted 

 by x, i/, z, the distance of this last point from the focus will be 

 the square root of the quantity 



(x - xtf + (y - ytf + s 2 ; 



and if a plane drawn through 8, parallel to the directive plane, 

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

 the square root of the quantity 



(x - xtf seo 2 + (y - y z }* ; 



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

 It is given by Pappus at the end of the Seventh Book of his Mathematical Collec- 

 tions. 



