368 THE FORMS OF CELLS [ch. 



and to the same conclusion we may also come by ways purely 

 mathematical. The plane and the sphere are two obvious examples 

 of such surfaces, for in both the radius of curvature is everywhere 

 constant. 



From the fact that we may extend a soap-film across any ring 

 of wire, however fantastically the wire be bent, we see that there 

 is no end to the number of surfaces of minimal area which may be 

 constructed or imagined*. While some of these are very com- 

 phcated indeed, others, such as a spiral or helicoid screw, are 

 relatively simple. But if we Umit ourselves to surfaces of revolution 

 (that is to say, to surfaces symmetrical about an axis), w^e find, as 

 Plateau was the first to shew, that those which meet the case are 

 few in number. They are six in all, nam-ely the plane, the sphere, 

 the cyhnder, the catenoid, the unduloid, and a curious surface which 

 Plateau called the nodoid. 



A B CD 



Fig. 104. Roulettes of the conic sections. 



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 Delaunayf, in 1841) 

 that the plane curves by whose revolution they are generated are 

 themselves generated as "roulettes" of the conic sections. 



Let us imagine a straight line, or axis, on which a circle, ellipse or 

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

 a line in some relation to the fixed axis, and this fine (or roulette), 

 when we rotate it around the axis, will describe in space one or 

 another of the six surfaces of revolution of which we are speaking. 



If we imagine an elhpse so to roll on a base-hne, either of its foci 

 will describe a sinuous or wavy fine (Fig. 104, B) at a distance 



* To fit a minimal surface to the boundary of any given closed curve in space is 

 a problem formulated by Lagrange, and commonly known as the "problem of 

 Plateau," who solved it with his soap-films. 



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

 de M. Liouville, vi, p. 309, 1841. Cf. {int. al.) J. Clerk Maxwell, On the theory of 

 rolUng curves. Trans. R.S.E. xvi, pp. 519-540, 1849; J. K. Wittemore, Minimal 

 surfaces of rotation, Ann. Math. (2), xix, 1917, Amer. Journ. Math, xl, p. 69, 

 1918; Crino Loria, Courbes planes speciales, theorie ef histoire, Milan, 574 pp., 1930. 



