449 



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-x,f + {y-y{f +z'=m'{{x-x^f^ec''<^ + {y-y^n, (1) 

 which, by making 



K-=z l—m^ sec^^, B :=: 1 — m^, 



G = ?ra^a?2 sec'^^ — a?i, h = m^yi—yi, (2) 



K = n^ixi sec^<j> + y.2^) — xi^ — y\, 



may be put under the form 



Ax^ + B?/2 + z''+2gx + 2ny = K, (3) 



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

 the surface, and that the planes of X'% 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 — ^, or into 180° — ^, the directive plane may have two 

 positions equally inclined to the plane of xy, and therefore 

 equally inclined to the directrix. Indeed it is obvious 

 that, if through the point S we draw two planes making 

 equal angles with the directrix, and cutting it in the points 

 D and D' respectively, the distances SD 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 y, 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 dis- 

 tance SD from the point D ; and, as the locus of a point 



