26 j\I, J. Plateau on the Equilibrium-figures of Revolution 



this experiment, however, is not so easily made to succeed as the 

 first. 



It must be observed that this experiment, as well as all the 

 following ones, must alike be made in the alcoholic mixtui-e, and 

 that the precautions, indicated in the first and second series*, for 

 rt;ducing the two liquids to the same density, and causing them 

 to exercise no sensible chemical action upon one another, must 

 be attended to. 



Since the complete meridian curve cannot terminate as above 

 described, the preceding figure is only a partial one. I follow 

 the meridian curve, therefore, beyond the extremities of this 

 same figure. 



In the first place, these extreme points cannot be points of 

 inflection ; for if they were, the radius of curvature would 

 be there infinite, and the equation of equilibrium reduced to 



r^, = C. But we have seen that, between its vertex and the 



N 



points under consideration, the curve presents two points of in- 

 flection, at which, therefore, the equation really reduces itself to 



the form ^^ = C. Hence the normals at the two former points 



must be equal to those at the two latter, which is evidently im- 

 possible according to the respective positions of these two pairs 

 of points, and the directions of the corresponding normals. I 

 have, it is true, here supposed that at a point of inflection the 

 radius of curvature is necessarily infinite, whereas it is well 

 known that the same may also be zero. In our meridian curves, 

 however, the vanishing of the radius of curvature is inadmissible, 



for the quantity ttf + i^ would be thereby rendered infinite at 



such a point. Beyond the extremities of the partial figure, 

 therefore, our curve still preserves a curvature of the same kind 

 as it had before reaching the same, that is to say it is still con- 

 cave towards the exterior, and the points at these extremities 

 arc at a minimum distance fi-om the axis. 



This being established, I set off, from one of these minimum 

 points and upon the production of the curve, an arc so small 

 that throughout its whole length the curvature either continually 

 increases or diminishes, and, from the same point but on the 

 other, or near side of the same, I set ofi" an are of equal length. 

 At all points of these two small arcs the radius of curvature 

 and the normal have opposite directions, so that the quantity 



»T+ ?^ constitutes a difl'erence, and consequently cannot remain 

 * Taylor's Scientific Memoirs, vols. iv. and v. 



