368 On Figures of Equilibrium of a Liquid Mass without Weight. 



One point still remained to be cleared up. According to the 

 calculation I have been speaking of, in order that the surface may 

 be diminished by the kind of transformation here indicated, it is 

 sufficient that the sum of the lengths of an expanded and con- 

 tracted part of the figure should exceed the circumference of the 

 cylinder ; and this allows us to attribute to this sum an infinite 

 number of different values. But nevertheless, as has been shown 

 in the Second Series, in a cylinder that is very long relatively to 

 its diameter, when the transformation takes place quite regularly, 

 the sum in question is always the same for the same cylinder 

 under the same circumstances, whence we must infer that there 

 is some special condition which regulates the choice of the mass. 

 We may add that analogous considerations apply to the other 

 unstable figures. I examine the matter, and arrive at the fol- 

 lowing as a very probable conclusion : — that, among all possible 

 changes of shape which would diminish the surface, the mole- 

 cular forces choose that one which allows the smallest possible 

 departure of the mass from another figure of equilibrium. In 

 the cylinder, for example, the mass will assume at the beginning 

 of the transformation the figure which, considering the resist- 

 ances, makes the nearest possible approach to. the unduloid. 



The combination of liquid films, which I have studied parti- 

 cularly in my Sixth Series, likewise presents some remarkable 

 phenomena in relation to their stability. I tried to establish ex- 

 perimentally that every equilibrated system of films in which 

 more than three films meet at one liquid edge, or more than four 

 liquid edges at one liquid point, is an unstable system. M. La- 

 marle has since discussed the question in detail in his memoir 

 " On the Stability of Liquid Systems formed of thin Films." Set- 

 ting out from the general principle that I had laid down at the 

 end of my Sixth Series — namely, that in every permanent assem- 

 blage of films the sum of the areas of the films must be a mini- 

 mum, — he succeeds in giving a strict demonstration of the condi- 

 tions above mentioned relatively to the number of films and of 

 liquid edges ; and he arrives besides at many other interesting 

 results. 



Lastly, I recall the fact that M. Duprez, in his memoir " On 

 a particular case of the Equilibrium of Liquids," has investigated 

 a phenomenon in which the stability or instability of a liquid 

 surface depends conjointly on the action of gravity and of mole- 

 cular forces. The phenomenon referred to is the familiar one of 

 the suspension of a liquid in a vertical tube open at the lower 

 end when the diameter of the opening is below a certain limit ; 

 but M. Duprez has shown that this limit is much greater than 

 was generally supposed. He has held up water in this way in a 

 tube the diameter of whose opening was 19*85 millims. ; while 



