476 



hyperboloid is similar to a parallel section of the asymptotic 

 cone, and when the latter section is a hyperbola its asymp- 

 totes are parallel to two sides of the cone. 



PART ri. — PROPERTIES OF SURFACES OF THE SECOND ORDER. 



§ 1. In the preceding part of this paper it has been ne- 

 cessary to enter into details for the purpose of communicat- 

 ing fundamental notions clearly. In the following part, 

 which will contain certain properties of surfaces of the se- 

 cond order, we shall be as brief as possible; giving demon- 

 strations of the more elementary theorems, but confining 

 ourselves to a short statement of the rest. 



Many consequences follow from the principles already 

 laid down. 



Through any directrix of a surface of the second order 

 let a fixed plane be drawn cutting the surface, and let S be 

 any point of the section. If the directrix and its focus F be 

 modular, and if a plane always parallel to the same directive 

 plane be conceived to pass through S and to cut the direc- 

 trix in D, the directive distance SD will be always parallel to 

 a given right line, and will therefore be in a constant ratio to 

 the perpendicular distance of S from the directrix. This 

 perpendicular distance will consequently bear a given ratio 

 to SF, the distance of the point S from the focus. And the 

 same thing will be true when the directrix and focus are 

 umbilicar, because the perpendicular distance of the point 

 S from the directrix will be in a constant ratio to its distance 

 from each directive plane drawn through the directrix. 



The fixed plane of section will in general contain another 

 directrix parallel to the former, and belonging to the same 

 focal ; and it is evident that the perpendicular distance of S 

 from this other directrix will be in a given ratio to its dis- 

 tance SF' from the corresponding focus F', the ratio being 

 the same as in the former case. Hence, according as the 

 point S lies between the two directrices, or at the same side 



