V] OF MINIMAL SURFACES 



alternately maximal and minimal from the axis; this wavy Hne, 

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

 surface which we call the unduloid, and the more unequal the two 

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

 sinuosity of the roulette. If the two axes be equal, then our elhpse 

 becomes a circle ; the path described by its roUing centre is a straight 

 line parallel to the axis (A), and the sohd of revolution generated 

 therefrom will be a cylinder: in other words, the cylinder is a 

 "Hmiting case" of the unduloid. If one axis of our ellipse vanish, 

 while the other remains of finite length, then the elhpse is reduced 

 to a straight line with its foci at the two ends, 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 ellipse vanish, but the other be infinitely 

 long, then the roulette described by the focus of this ellipse will be 

 a circular arc at an infinite distance; i.e. it will be a straight line 

 normal to the axis, and the surface of revolution traced by this 

 straight hne turning about the axis will be a plane. If we imagine 

 one focus of our ellipse to remain at a given distance from the axis, 

 but the other to become infinitely remote, that is tantamount to 

 saying that the elhpse becomes transformed into a parabola; and 

 by the rolling of this curve along the axis there is described a 

 catenary (D), whose solid of revolution is the catenoid. 



Lastly, but this is more difficult to imagine, we have the case of 

 the hyperbola. We cannot well imagine the hyperbola rolling upon 

 a fixed straight line so that its focus shall describe a continuous 

 curve. But let us suppose that the fixed hne is, to begin with, 

 asymptotic to one branch of the hyperbola, and that the rolUng 

 proceeds until the hne is now asymptotic to the other branch, that 

 is to say touching it at an infinite distance; there will then be 

 mathematical continuity if we recommence rolling with this second 

 branch, and so in turn with the other, when each has run its course. 

 We shall see, on reflection, that the line traced by one and the 

 same focus will be an "elastic curve," describing a succession of 

 kinks or knots (E), and the solid of revolution described by this 

 meridional line about the axis is the so-called nodoid. 

 I 



The physical transition of one of these surfaces into another can 



