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 dirigent 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 

 elHpse 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 m is less than cos f, 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 1— b are 

 always positive, the focal and dirigent curves are ellipses. 



