On the Surfaces of the Second Order. 285 



varieties, which are contained in the general equation of the 

 second degree, but are excluded from that method of genera- 

 tion by reason of the simplicity of their forms namely, the 

 sphere, the right cylinder on a circular 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 spheroid, 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.! In point 

 of simplicity, 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 

 were, in fact, studied by the Greek geometers, $ and, along with 

 the oblate spheroid and the cone, they make up all the surfaces 

 of the second order with which the ancients were acquainted. 

 Except the cone, the surfaces considered by them are all of 

 revolution; and there is only one surface of revolution, the 

 hyperboloid of one sheet, which was not noticed until modern 

 times. This surface is mentioned (under the name of the 

 hyperbolic cylindroid) by Wren, who remarks that it can 

 be generated by the revolution of a right line round another 

 right line not in the same plane. As to the general conception 

 of surfaces of the second order, the suggestion of it was reserved 

 for the algebraic geometry of Descartes. In that geometry the 



* The case of two parallel planes is also excluded, but 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. 



t A Paper by M. Chasles, on these surfaces of revolution, will be found in the 

 "Memoirs/' of the Academy of Brussels, torn. v. (An. 1829). 



J The hyperboloid of two sheets, and the paraboloid of revolution, were known 

 by the name of conoids. Archimedes has left a treatise on Conoids and Spheroids, 

 as well as a treatise on the Sphere and Cylinder. 



In the Philosophical Transactions for the year 1669, p. 961. 



