which are neither Spherical nor Cylindrical. 33 



attribute a constant value to the distance between the vertex of 

 a loop and the axis of revolution, these loops become elongated 

 whilst they recede from each other ; so that if we consider any 

 one in particular and pass to the limit, all the rest, together 

 with the intermediate arcs, are transported to infinity, or, in 

 other words, disappear; and this one beomes transformed into a 

 curve with two infinite branches, which is no other than the 

 catenary placed in the manner before described; hence the 

 limit of this third mode of variation is the catcnoid. Thus the 

 catenoid, itself one of the limits of the unduloid, forms a second 

 phase of transition between the unduloid and the nodoid. 



The experiments indicated in this abstract, together with 

 others described in the memoir, exhaust all the cases of realiza- 

 tion between two equal rings or discs ; if there is still another 

 equilibrium-figure, therefore, it must be such that isolated por- 

 tions of the same cannot be obtained under these conditions ; 

 from which fact we may conclude that its meridian curve pre- 

 sents no maxima or minima of distance from the axis. More- 

 over, since the meridian curve cannot cut the axis, it must com- 

 mence from a point at infinity upon an asymptote parallel to the 

 axis, and afterwards recede continually from the latter until it 

 reaches a second point also at infinity. This granted, it may 

 be observed that, at the first of these two extreme points, the 

 radius of curvature is necessarily infinite and the normal finite, 



80 that the equation of equilibrium reduces itself to 5t = C. 

 Now if the curve had anywhere a point of inflection, this equa- 

 tion would again be reduced to at = C, and consequently the 



normals corresponding, respectively, to this and to the first 

 extreme point would be equal, which is clearly impossible. The 

 curve, therefore, if it exists, must everywhere turn its convexity 

 towards the axis, and hence we see without difficulty that its 

 second extreme point will be infinitely distant from this axis, 

 and that the normal will be there infinite. But, evidently, the 

 radius of curvature will also be infinite at this second extreme 



point, so that the quantity ^ + |^- will be there equal to zero, 



which is impossible, since at the first extreme point this quantity 

 has a finite value. The condition of equilibrium, thercrorc, 

 cannot be satisfied. 



Thus the only equilibrium-figures of revolution are the sphere, 

 the plane, the cylinder, the unduloid, the catenoid and the 

 nodoid. 



Phil. Mug. S. 4. Vol. 1(). No. lot. Jahj 1858. D 



