24 PROTOPLASM 



be fulfilled, with an incHnation of 116° 33' 54" formed by 

 the surfaces meeting at one edge, and an angle of 108° which 

 the edges, that meet at one corner, form one with another. 

 A soap lather formed of bubbles as equal in size as pos- 

 sible, visibly consists mainly of dodecahedric alveoli. They 

 naturally can never, however, be regular dodecahedra, since 

 the figure is one that does not quite satisfy the conditions to 

 be fulfilled. Since, however, we are deahng with fluid 

 lamellae, this deficiency in the angles can easily be com- 

 pensated, and the condition of equilibrium established, if the 

 lamellae equalise the difference in the angles of contact, by 

 curving slightly towards the edges at which they meet one 

 another. As a natural consequence, only the angles formed 

 by the tangential planes of the curved surfaces at their edge 

 of contact amount to 120°. That such curvings of the 

 lamellae frequently occur in macroscopic froths, so as to 

 estabhsh conditions of equiKbrium, is shown at once by 

 observation ; similarly the edges are also frequently curved 

 in the same manner, in order to comply with the condition 

 that they should form at the nodal points angles of 109° 

 28' 16" with adjoining edges. 



Considering what has been stated, it will readily be 

 imderstood that the alveoli of froth may form polyhedra of 

 the most various kinds, from tetrahedra to those enclosed by 

 the highest possible number of sides, and therefore that the 

 image of our microscopic froth will show polygons of very 

 different kinds. Nevertheless one condition [wiU always 

 obtain, that in a nodal point or an edge only three lines 

 meet one another, which may, however, enclose very variable 

 angles. The latter point depends, in the first place, on the 

 fact that in such fine microscopical froths as are under 

 consideration here, in which the breadth of the meshes 

 remains, as a rule, under '001 mm., it is not possible to dis- 

 tinguish in the microscopic image whether the three lines 

 radiating from a nodal point are the sections of three 

 lamellae meeting one another at one edge, or whether the 

 union of three edges is before us. Further, the section of 

 the plane of the image may pass through the edge at which 

 the lamellae meet one another in any sort of way, or rather 



