378 THE FORMS OF CELLS [ch. 



cylinder l/R' =- 0, and in the sphere 1/r = 1/r'. Therefore our 

 relation of equality becomes l/R = 2/r, or r = 2R; which means 

 that the sphere in question has just twice the radius of the cylinder 

 of which it forms a cap. 



And if Ob, the radius of the sphere, be equal to twice the radius 

 (Oa) of the cyhnder, it follows that the angle aOb is an angle of 60°, 

 and bOc is also an angle of 60° ; that is to say, the arc be is equal to 

 Jtt. In other words, the spherical disc which (under the given 

 conditions) caps our cylinder is not a portion taken at haphazard, 

 but is neither more nor less than that portion of a sphere which is 

 subtended by a cone of 60°. Moreover, it is plain that the height 

 of the spherical cap, de, = Ob — ab = R (2 — ^/3) = 0'27R, where 

 R is the radius of our cylinder, or one-half the radius of our spherical 

 cap: in other words the normal height of the spherical cap over 

 the end of the cylindrical cell is just a very little more than one- 

 eighth of the diameter of the cylinder, or of the radius of the sphere. 

 And these are the proportions which we recognise, more or less, 

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

 of Spirogyra, when one end is free and capped by a portion of a 

 sphere*. 



Among the many 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 stabihty; only the plane and the 

 sphere, or any portion of a sphere, are perfectly stable, because 

 they are perfectly symmetrical, figures. 



Perhaps it were better to say that their symmetry is such that 

 any small disturbance will probably readjust itself, and leave the 

 plane or spherical surface as it was before, while in the other 

 configurations the chances are that a disturbance once set up will 

 travel in one direction or another, increasing as it goes. For 

 equihbrium and probabihty (as Boltzman told us) are nearly alHed: 



* The conditions of stability of the cyhnder, and also of the catenoid, are 

 explained with the utmost simplicity by Clerk Maxwell, in his article, already 

 quoted, on "Capillarity.'' On the catenoids, see A. Terquem. C.R. xcii, pp. 407-9, 

 1881. 



