286 On the Surfaces of the Second Order. 



curves previously known as sections of the cone are all expressed 

 by the general equation of the second degree between two co- 

 ordinates ; and hence it occurred to Euler* about a century ago, 

 to examine and classify the different kinds of surfaces comprised 

 in the general equation of the second degree among three co- 

 ordinates. The new and more general forms thus brought to 

 light have since engaged a large share of the attention of geo- 

 meters ; but the want of some other than an algebraic principle 

 of connexion has prevented any great progress from being made 

 in the investigation of such of their properties as do not im- 

 mediately depend on transformations of co-ordinates. This 

 want the modular method of generation perfectly supplies, by 

 evolving the different forms from a simple geometrical concep- 

 tion, at the same time that it brings them within the range of 

 ideas familiar to the ancient geometry, and places their relation 

 to the conic sections in a striking point of view. 



It may be well to remark that the excepted surfaces are 

 limits of surfaces which can be generated modularly, as the 

 circle is the limit of the ellipse in the analogous generation of 

 the conic sections. Thus the sphere is the limit of an oblate 

 spheroid, one of whose axes remains constant, while its focal 

 circle is indefinitely diminished ; and the right circular cylinder 

 is the limit of an elliptic cylinder, whose focal lines are con- 

 ceived to approach indefinitely to coincidence with each other 

 and with the axis of the cylinder, while one of the axes of the 

 principal elliptic section remains constant. In these cases the 

 dirigent lines, along with the directrices, move off to infinity. 

 The other three excepted surfaces correspond to the supposition 

 = 90, which was excluded in the discussion of the general 

 equation (1). For if we make m sec = , the quantity which 

 constitutes the right-hand member of that equation may be 

 written 



n z (x - x 2 y + n? (y - y^y cos 2 ; 



and if we suppose n to remain finite and constant, while $ 



* See his Introduetio in Analysin Iiifinitorum, p. 373. Lausanne, 1748. 



