454 



called, from its employment of the modulus. A focal curve 

 which can be so used shall be distinguished as a modular fo- 

 cal ; but each focal, whether modular or not, shall be sup- 

 posed to have a dirigcnt curve and a dirigent cylinder con- 

 nected with it by the relations already laid down. 



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 sur- 

 face. 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 ver- 

 tex of the cone, and each hyperbola is reduced to the pair of 

 right lines which were previously the asymptotes. The diri- 

 gent cylinder, in the one case, is narrowed into a right line ; 

 in the other case it is converted into a pair of planes, which 

 we may call the dirigent planes of the cone. 



§ 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 ?w is less than cos 0, the quantities a, b, k, p, q, r 

 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 I — b are 

 always positive, the focal and dirigent curves are ellipses. 



