V] 



OF FIGURES OF EQUILIBRIUM 



227 



sphere. And these are the proportions which we recognise, ander 

 normal circumstances, in such a case as the cylindrical cell of 

 Spirogyra where its free end is capped by a portion of a sphere. 



Among the many important theoretical discoveries which we 

 owe to Plateau, one to which we have just referred is of peculiar 

 importance : namely that, with the exception of the sphere and 

 the plane, the surfaces with which we have been dealing are only 

 in complete equilibrium within certain dimensional limits, or in 

 other words, have a certain definite limit of stability ; only the plane 

 and the sphere, or any portions of a sphere, are perfectly stable, 

 because they are perfectly symmetrical, figures. For experimental 

 demonstration, the case of the cylinder is the simplest. If we 

 produce a liquid film having the form of a cylinder, either by 



Fig. 68. 



drawing out a bubble or by supporting between two rings a 

 globule of oil, the experiment proceeds easily until the length of 

 the cylinder becomes just about three times as great as its diameter. 

 But somewhere about this limit the cylinder alters its form ; it 

 begins to narrow at the waist, so passing into an unduloid, and 

 the deformation progresses quickly until at last our cylinder 

 breaks in two, and its two halves assume a spherical form. It is 

 found, by theoretical considerations, that the precise limit of 

 stability is at the point when the length of the cylinder is exactly 

 equal to its circumference, that is to say, when L = IttR, or when 

 the ratio of length to diameter is represented by tt. 



In the case of the catenoid. Plateau's experimental procedure 

 was as follows. To support his globule of oil (in, as usual, a 

 mixture of alcohol and water of its own specific gravity), he used 



15—2 . 



