472 



order, and the right line AF is a normal at F to the sur- 

 face which is the locus of this point. Moreover, if through 

 the point A three or more planes be drawn parallel to fixed 

 planes, and perpendiculars be dropped upon them from 

 any point S whose coordinates are x, ?y, s, the right-hand 

 member of the equation (38) may be conceived to represent 

 a given homogeneous function of the second degree of these 

 perpendiculars ; and the given surface may therefore be 

 regarded as the locus of a point S, such that the square 

 of the distance SF is always equal to that function. 



§11. In the enumeration of the surfaces capable of being 

 generated by the modular method, we miss the five follow- 

 ing varieties, which are contained in the general equa- 

 tion of the second degree, but are excluded from that me- 

 thod of generation by reason, of the simplicity of their 

 forms — namely, the sphere, the right cylinder on a circu- 

 lar base, and the three surfaces which may be produced 

 by the revolution of a conic section (not a circle) round 

 its primary axis.* These three surfaces are the prolate sphe- 

 roid, the hyperboloid of two sheets, and the paraboloid 

 of revolution ; and the circumstance that the foci of the 

 generating curves are also foci of the surfaces, renders it 

 easy to investigate their focal properties.f In point of sim- 

 plicity, the excepted surfaces are to the other surfaces of 

 the second order what the circle is to the other conic sections, 

 the circle being, in like manner, excepted from the curves 

 which can be generated by the analogous method in piano; and 

 the geometry of the five excepted surfaces may therefore be 

 regarded as comparatively elementary. These five surfaces 



* The case of two parallel planes is also excluded, hut it is not here taken 

 into account. The case of two parallel right lines is in like manner excluded 

 from the corresponding generation of lines of the second order. 



f A paper hy M. Chasles, on these surfaces o^ revolution, will be found in 

 the Memoirs of the Academy of Urussels, torn. v. (An. 1829). 



