450 



whose distances from the two points F and D are in a con- 

 stant 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 <f), or by changing ^ into its supplement, we may sup- 

 pose this angle (when it is not zero) to be always positive 

 and less than 90° ; for the supposition <(> =z 90° is to be ex- 

 cluded, as it would make the secant of <J) 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 surfaces. The first case, when 

 neither a nor b vanishes, gives the ellipsoid, the two hyper- 

 boloids, and the cone ; the second, when either or each of 

 these quantities is zero, includes the two paraboloids and 

 the difi'erent kinds of cylinders. 



§ 2. First Class of Sin faces. — When neither a nor b va- 

 nishes, we may make both g and H vanish, by properly as- 

 suming the origin of coordinates. Supposing this done, we 



have 



xi = m^X2 sec^tp, i/i zz rr^yi, (4) 



the equation of the surface being then 



kX^ -h B^^ -h S^ = K, (5) 



in which the axes of coordinates 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 coordinates; when 

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



Eliminating x^-, y-^ from the value of k, by means of the 

 relations (4), we get 



l-^-'^ + i 



K = .— .r/ + -— y.^ (6) 



