180 SURFACES AND MEMBRANES 



made up of two parts: the area of the curved surface of the spherical 

 segment with surface energy equal to (irh 2 + ira 2 )Tcp, and the circular 

 interface with surface energy equal to Ta 2 (T GC — T GP ) ergs per square 

 centimeter. Then the total energy is 



E = T(h 2 + a 2 )T CP + *a 2 (T 0C - T GP ) 



where T represents the tensions designated by the three different inter- 

 facial subscripts. 



Let Tcp — n and T GC ~ Top — w. The above equation may then 

 be rewritten as 



E = nir(h 2 + a 2 ) + rrnra 2 

 where, because the volume of the cell remains constant, 



3/i 3 



a 2 



In the process of spreading, the volume of the cell is assumed to 

 remain constant, so that the problem reduces itself to ascertaining the 

 magnitude of the height h of the cell at equilibrium for given values of 

 cell, surface, and plasma characteristics, i.e., for values of m and n or 

 values of m/n. 



Substituting for a its values in terms of h in the above equation for E, 

 and differentiating the result with respect to h, gives 



,a[ 2n _ |(m + n) _8 f(m + n) ] 



dE_ 



dh ' 3/i 



2 



Since at equilibrium the surface energy must be a minimum, i.e., 



4(w ~\~ 7l) 



dE/dh = 0, it follows that h 3 = — . Geometrically it can be 



2n — m 



seen from Fig. V-2 that 



4 - 2h 3 



cos 6 = 3 



4 + h 6 



Substituting the above value of h z in this relation, we find that 



m T GP - T GC 



cos 6 = = 



n T 



CP 



It is possible to use this expression to predict what will happen when a 

 cell of uniform density, immersed in plasma of the same density, makes 

 contact with a horizontal surface. Suppose that it spreads until its 

 contact angle is 90°. What is the height of the cell under these circum- 



