II. i CELL-DIVISION 43 



drops in which the principle of least surfaces is obeyed. Soap- 

 bubbles form similar systems. But where this condition is not 

 fulfilled, as in oil-drops floated, for instance, between alcohol and 

 water, the drops either unite or separate, each retaining 1 its 

 spherical form. 



The geometrical analysis of such systems given by Plateau 



is as follows. In a system of two bubbles the curvature of the 



/ 



surface of contact is given by the equation r = ,, where r is 



P-P 

 the radius of that surface, p, p the radii of the larger and 



smaller bubbles. Since the pressure varies inversely with the 

 radius, the surface of contact is convex towards the larger 

 bubble. When p p , r = a, and this surface is plane. Since 

 there is equilibrium the external surfaces of the bubbles and 

 their common surface meet at angles of 120. 



In a system of three bubbles there are three contact surfaces ; 

 these meet in one line and make angles of 120 with one 

 another. When there are four bubbles, however, the four con- 

 tact surfaces cannot meet in one line except for an inappreciable 

 instant; they immediately shift their positions in such a way 

 that two opposite bubbles meet and separate the other two from 

 one another. There are thus five surfaces of contact, and these 

 make angles of 120 with one another as before. This is the 

 arrangement when four bubbles whether equal or unequal is no 

 matter are placed side by side in the same plane. When, 

 however, one bubble is placed in a different plane to the re- 

 maining three, four surfaces are formed and disposed in such 

 a manner that the four lines, each formed by the intersection 

 of three of these surfaces, meet in one point, making with one 

 another angles of 109 28' 16", the angles at the centre of a 

 tetrahedron. In short, the four are now tetrahedrally arranged. 

 The systems of drops of jelly alluded to above arrange them- 

 selves as do soap-bubbles under similar circumstances. What 

 holds good of four holds good of an assemblage of any number 

 of bubbles. The size of the bubbles is a matter of indifference, 

 except to the curvature of the surfaces of contact, and, to 

 a certain extent, to the arrangement. Thus, if four equal 

 bubbles be placed in a plane, they will form together five 

 surfaces of contact, one of which will be between two opposite 



