453 



and also that 



P, =P — R, Qi = Q — R. (13) 



From equations (12) we see that Pi and P2 have always the 

 same sign, as also Qi and Q2 ; and that, neglecting signs, the 

 semiaxes of the surface are mean proportionals between the 

 corresponding 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 recipro- 

 cal 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, r, 

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

 with relation to these magnitudes. Supposing, therefore, 

 the quantities 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 reference, shall be termed the se- 

 condary axis ; while the quantity which has an intermediate 

 algebraic value shall mark the middle or mean axis. Then, 

 since both p, and q, 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 there- 

 fore 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, 

 ifRbethe 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 

 determined, 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 



