360 Prof. J. Plateau on the Figures of Equilibrium 



without weight ; and he accordingly thus obtained the equation to 

 surfaces of constant mean curvature. Meusnier had pointed out 

 the skew helicoid with directing plane as a surface whose mean 

 curvature was nothing ; M. Catalan has proved that this helicoid 

 and the plane are the only ruled surfaces of the mean curvature 

 zero. M. Lamarle has integrated the general equation in the 

 case of helicoids, and has thus found four other surfaces besides 

 the skew helicoid with directing plane. Mr. Jellett has indicated 

 a simple condition which every closed surface of constant mean 

 curvature, except the sphere, must satisfy. 



The general case of surfaces whose mean curvature is zero was 

 first treated by Monge and Legendre ; M. Scherk has deduced, 

 from the integral which they gave, the equations in finite coor- 

 dinates of five particular surfaces. M. Ossian Bonnet has made 

 known another more convenient integral, and has applied his 

 method to the investigation of surfaces of the kind in question 

 which pass through a determinate continuous contour. M. Ser- 

 ret has shown how they can be made to pass through a series of 

 straight lines not situated in the same plane ; and M. Mathet, 

 by a method different from that of M. Bonnet, how they can be 

 made to pass through a given plane curve. M. Catalan has 

 published another integral still of the general equation, and has 

 deduced from it several surfaces. 



I also recall the researches of Dupre, Rennes, and M. Van der 

 Mensbrugghe relating especially to the tension of liquid sur- 

 faces, researches of which I have spoken already in the Eighth 

 Series. 



Now let us see what are the experimental verifications. I 

 have measured the limit of stability of the catenoid by means of 

 a laminar catenoid formed between two equal rings, whose dis- 

 tance could be gradually varied and exactly measured ; and the 

 result was found to agree perfectly with that deduced from Gold- 

 schmidt's calculation. 



I have applied the general principle which concludes my 

 Seventh Series to the realization of the skew helicoid with direct- 

 ing plane, by employing a closed outline of iron wire composed 

 of two spires of a regular helix, of a part of the axis, and of two 

 straight lines connecting this part with the extremities of the 

 helix. When this outline is taken out of the glycerine-liquid, it 

 is found to be occupied by a beautiful curved film which repre- 

 sents exactly the helicoid in question. 



I have realized in the same way in the laminar form, upon an 

 appropriate framework, a portion of a remarkable surface first 

 discovered by M. Scherk, and since discussed by M. Catalan. 



M.Van der Mensbrugghe, by applying still the same principle, 

 has also realized another of M. Scherk's surfaces. 



