On the Surfaces of the Second Order. 267 



From equations (12) we see that PI and P 2 have always the same 

 sign, as also Qi and Q 2 ; and that, neglecting signs, the semi- 

 axes of the surface are mean proportionals between the corre- 

 sponding semiaxes of the focal and dirigent curves. These curves 

 are therefore reciprocal polars with respect to the section made 

 in the surface by the plane of xy ; and it would be easy to show 

 that the points F and A are reciprocal points, or that a tangent 

 applied at one of them to the curve which is its locus has the 

 other for its pole. 



The focal curve, when we know in which of the principal 

 planes it lies, is determined by the conditions (13) ; and as it 

 depends . on the relative magnitudes of the quantities P, Q, It, 

 it will be convenient to distinguish the axes of the surface, with 

 relation to these magnitudes. Supposing, therefore, the quan- 

 tities P, Q, R to be taken with their proper signs, as they are 

 in the equation (9), that axis to which the greatest of them 

 (which is always positive) refers shall be called the primary 

 axis ; and that to which the quantity algebraically least has re- 

 ference shall be termed the secondary axis ; while the quantity 

 which has an intermediate algebraic value shall mark the middle 

 or mean axis. Then, since Pi and Qi will be negative, if R be 

 the greatest of the quantities aforesaid, the focal curve cannot 

 lie in the plane of the mean and secondary axes. Its plane 

 must therefore pass through the primary axis: it will be the 

 plane of the primary and mean axes, if R be the least of the 

 three quantities ; but the plane of the primary and secondary 

 axes, if R be the intermediate quantity. In the former case the 

 curve will be an ellipse, in the latter a hyperbola ; and we shall 

 extend the name of focal curves to both the curves so deter- 

 mined, though it may happen that only one of them can be 

 used in the generation of the surface by the modular method, as 

 the method of which we are treating may be called, from its 

 employment of the modulus. A focal curve which can be so 

 used shall be distinguished as a modular focal; but each focal, 

 whether modular or not, shall be supposed to have a dirigent 

 curve and a dirigent cylinder connected with it by the relations 



