486 THE FORMS OF TISSUES [ch. 



triple, instead of one quadruple, conjunction. In like manner, when 

 four billiard-balls are packed close upon a table, two tend to come 

 together and separate the other two. 



Let us epitomise the Law of Minimal Areas and its chief clauses 

 or corollaries in the particular case of an assemblage of fluid films, 

 as was first done by Lamarle*. Firstly and in general: In every 

 liquid system of thin films in stable equihbrium, the sum of the 

 areas of the films is a minimum. From, observation and experience, 

 rather than by demonstration, it follows that (2) the area of each 

 is a minimum under its own limiting conditions; and further that 

 (3) the mean curvature of any film is constant throughout its whole 

 area, null when the pressures are equal on either side and in other 



Fig. 172. A, an unstable arrangement of four cells or bubbles. B, the normal and 

 stable configuration, showing the polar furrow. 



cases proportional to their difference. Less obvious, very important, 

 and hkewise subject (but none too easily) to rigorous mathematical 

 proof, are the next two propositions, both of which had been laid 

 down empirically by Plateau: (4) the films meeting in any one edge 

 are three in number; (5) the crests or edges meetirg in any one 

 corner are four in number, neither more nor less. Lastly, and 

 following easily from these: (6) the three films meeting in a crest 

 or edge do so. at co-equal angles, and the same is true of the four 

 edges meeting in a corner. 



Wherever we have a true cellular complex, an arrangement of 

 cells in actual physical contact by means of their intervening 

 boundary walls, we find these general principles in force; we must 

 only bear in mind that, for their easy and perfect recognition, we 



* Ernest Lamarle, Sur la stabilite des systemes liquides en lames minces, Mem. 

 de VAcad. R. de Belgique, xxxv, xxxvr, 1864-67. 



