264 On the Surfaces of the Second Order. 



from the point D ; and, as the locus of a point whose distances 

 from the two points .Fand D are in a constant ratio to each 

 other is a plane or a sphere, according as the ratio is one of 

 equality or not, it follows that the section aforesaid will be a 

 right line in the one case, and a circle in the other. Hence 

 it appears that all directive sections, that is, all sections made in 

 the surface by planes parallel to either of the directive planes, 

 are right lines when the modulus is unity, and circles when the 

 modulus is different from unity. 



Since the equation (3) is not altered by changing the sign of 

 0, or by changing $ into its supplement, we may suppose this 

 angle (when it is not zero) to be always positive and less than 

 90 ; for the supposition = 90 is to be excluded, as it would 

 make the secant of $ infinite, and the directive planes parallel 

 to the directrix. In the discussion of the equation there are two 

 leading cases to be considered, answering to two classes of sur- 

 faces. The first case, when neither A nor B vanishes, gives the 

 ellipsoid, the two hyperboloids, and the cone ; the second, when 

 either or each of these quantities is zero, includes the two para- 

 boloids and the different kinds of cylinders. 



2. First Class of Surfaces. When neither A nor B vanishes, 

 we may make both G and H vanish, by properly assuming the 

 origin of co-ordinates. Supposing this done, we have 



sec 2 ^, yi = nfyz, . (4) 



the equation of the surface being then 



Ax 2 + By* + z* = K, (5) 



in which the axes of co-ordinates are of course the axes of the 

 surface. When K is not zero, the surface is an ellipsoid or hy- 

 perboloid, having its centre at the origin of co-ordinates ; when 

 K = 0, the surface is a cone having its vertex at the origin. 



Eliminating # 2 , y* from the value of K y by means of the re- 

 lations (4), we get 



