226 



THE FORMS OF CELLS 



[CH. 



breaks down. Moreover in our composite surfaces, when the 

 cylinder for instance is capped by two spherical cups or lenticular 

 discs, there is a well-defined ratio which regulates their respective 

 curvatures, and therefore their respective dimensions. These two 

 matters we may deal with together. 



Let us imagine a liquid drop which by appropriate conditions 

 has been made to assume the form of a cylinder ; we have already 

 seen that its ends will be terminated by portions of spheres. 

 Since one and the same liquid film covers the sides and ends of 

 the drop (or since one and the same delicate membrane encloses 

 the sides and ends of the cell), we assume the surface-tension (T) 

 to be everywhere identical ; and it follows, since the internal 

 fluid-pressure is also everywhere identical, that the expression 

 {1/R + l/R') for the cylinder is equal to the corresponding expres- 

 sion, which we may call (1/r + 1/r'), in the case of the terminal 

 spheres. But in the cylinder 1/R' = 0, and in the sphere 1/r = 1/r'. 

 Therefore our relation of equality becomes 1/R = 2/r, or r = 2R; 

 that is to say, 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 

 (Ort) of the cylinder, 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 

 1 77". 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, 



Fig. 67. 



^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 



