218 THE FORMS OF CELLS [oh. 



surfaces of revolution (that is to say, to surfaces symmetrical about 

 an axis), we find, as Plateau was the first to shew, that those which 

 meet the case are very few in number. They are six in all, 

 namely the plane, the sphere, the cylinder, the catenoid, the 

 unduloid, and a curious surface which Plateau called the nodoid. 



These several surfaces are all closely related, and the passage 

 from one to another is generally easy. Their mathematical inter- 

 relation is expressed by the fact (first shewn by Delaunay*, in 1841) 

 that the plane curves by whose rotation they are generated are 

 themselves generated as "roulettes" of the conic sections. 



Let us imagine a straight line upon which a circle, an elHpse 

 or other conic section rolls ; the focus of the conic section will 

 describe a line in some relation to the fixed axis, and this line 

 (or roulette), rotating around the axis, will describe in space one or 

 other of the six surfaces of revolution with which we are dealing. 



If we imagine an ellipse so to roll over a line, either of its foci 

 will describe a sinuous or wavy line (Fig. 61 b) at a distance 



alternately maximal and minimal from the axis ; and this wavy 

 line, by rotation about the axis, becomes the meridional line of 

 the surface which we call the unduloid. The more unequal the 

 two axes are of our elhpse, the more pronounced will be the 

 sinuosity of the described roulette. If the two axes be equal, 

 then our elhpse becomes a circle, and the path described by its 

 rolhng centre is a straight line parallel to the axis (A) ; and 

 obviously the solid of revolution generated therefrom will be a 

 cylinder. If one axis of our ellipse vanish, while the other remain 

 of finite length, then the ellipse is reduced to a straight line, and 

 its roulette will appear as a succession of semicircles touching one 

 another upon the axis (C) ; the solid of revolution will be a series of 

 equal spheres. If as before one axis of the elhpse vanish, but the 

 other be infinitely long, then the curve described by the rotation 



* Sur la surface de revolution dont la courbure moyenne est constante, Journ. 

 de M. LiouvilU, vi, p. 309, 1841. 



