On the Surfaces of the Second Order. 263 



so that, m being the modulus, the locus of the point S will be a 

 surface of the second order, represented by the equation 



(x - xtf + (y - 2/0 2 + * 2 = w 3 [v - x*J sooty + (y - y 2 ) 2 } , (1) 

 which, by making 



A = I - m z sec 2 ^, B = 1 - m 2 , 



G = m*x* sec 2 ^ - x, y H = m^ - y l9 (2) 



K - m z (x* sec 2 + / 2 2 ) - x? - y?, 



may be put under the form 



Ax* + By* + s 2 + 2Gv +. 2JTy = JT, (3) 



showing that the plane of xy is one of the principal planes of 

 the surface, and that the planes of xz and yz are parallel to 

 principal planes. 



Before we proceed to discuss this equation it may be well to 

 observe, that as it remains the same when is changed into - 0, 

 or into 180 - 0, the directive plane may have two positions 

 equally inclined to the plane of xy, and therefore equally in- 

 clined to the directrix. Indeed it is obvious that, if through 

 the point 8 we draw two planes making equal angles with the 

 directrix, and cutting it in the points D and H respectively, the 

 distances &D and SD' will be equal. Every surface described in 

 this way has consequently two directive planes ; and as each of 

 these planes is parallel to the axis of /, their intersection is 

 always parallel to one of the axes of the surface. This axis 

 may therefore be called the directive axis. The directive planes 

 have a remarkable relation to the surface, as may be shown 

 in the following manner : 



Suppose a section of the surface to be made by a plane which 

 is parallel to one of the directive planes, and which cuts the 

 directrix in D ; then the distance of any point S of the section 

 from the focus F will have a constant ratio to its distance SD 



