268 On the Surfaces of the Second Order. 



Since PI - Qi = P - Q, the foci of a focal curve are the same 

 as those of the principal section in the plane of which it lies, 

 and they are therefore on the primary axis of the surface. It 

 will sometimes contribute to brevity of expression, if we also 

 give the name of primary to the major axis of an ellipse and 

 to the real axis of a hyperbola. We may then say that the 

 primary axes of the surface and of its two focal curves are 

 coincident in direction ; and that (as is evident) the foci of 

 either curve are the extremities of the primary axis of the 

 other. 



If K be supposed to approach gradually to zero, while A and 

 B remain constant, the focal and dirigent ellipses will gradually 

 contract, and the focal and dirigent hyperbolas will approach 

 to their asymptotes, which remain fixed. When K actually 

 vanishes, the surface becomes a cone ; the two ellipses are each 

 reduced to a point coinciding with the vertex of the cone, and 

 each hyperbola is reduced to the pair of right lines which were 

 previously the asymptotes. The dirigent cylinder, in the one 

 case, is narrowed into a right line ; in the other case it is con- 

 verted into a pair of planes, which we may call the dirigent 

 planes of the curve. 



4. We have now to show how the different kinds of sur- 

 faces belonging to the first class are produced, according to the 

 different values of the modulus and other constants concerned 

 in their generation. 



I. When m is less than cos 0, the quantities A,B,E,P, Q, JR 

 are all positive, and Q is intermediate in value between P and 

 R. The surface is therefore an ellipsoid, and its mean axis is 

 the directive. As the quantities 1 - A and 1 - B are always 

 positive, the focal and dirigent curves are ellipses. 



Here we cannot suppose K to vanish, as the surface would 

 then be reduced to a point. 



When = 0, that is, when the directive planes coincide with 

 each other, and therefore with a plane perpendicular to the 

 directrix, so that SD is the shortest distance of the point S from 

 the directrix, the surface is a spheroid produced by the revo- 



