318 THE FORMS OF TISSUES [ch. 



lies just three-quarters of the way between an apex, a, and a point, 

 J), which is the centre of gravity of the opposite base. Therefore 



op = oa/3 = od/3. 



Therefore cos dop = ^ , and cos aod = — ^ . 



That is to say, the angle aod is just, as nearly as possible, 

 109° 28' 16". This angle, then, of 109° 28' 16", or very nearly 

 109 degrees and a half, is the angle at which, in this and every 

 other solid system of liquid films, the edges of the partition-walls 

 meet one another at a point. It is the fundamental angle in the 

 sohd geometry of our systems, just as 120° was the fundamental 

 angle of symmetry so long as we considered only the plane pro- 

 jection, or plane section, of three films meeting in an edge. 



Out of these two angles, we may construct a great variety of 

 figures, plane and solid, which become all the more varied and 

 complex when, by considering the case of unequal as well as equal 

 cells, we admit curved (e.g. spherical) as well as plane boundary 

 surfaces. Let us consider some examples and illustrations of 

 these, beginning with those which we need only consider in reference 

 to a plane. 



Let us imagine a system of equal cylinders, or equal spheres, 

 in contact with one another in a plane, and represented in section 

 by the equal and contiguous circles of Fig. 121. I borrow my 

 figure, by the way, from an old Italian naturalist, Bonanni (a 

 contemporary of BorelH, of Ray and Willoughby and of Martin 

 Lister), who dealt with this matter in a book chiefly devoted to 

 molluscan shells*. 



It is obvious, as a simple geometrical fact, that each of these 

 equal circles is in contact with six surrounding circles. Imagine 

 now that the whole system comes under some uniform stress. 

 It may be of uniform surface tension at the boundaries of all the 

 cells ; it may be of pressure caused by uniform growth or expansion 

 within the cells; or it may be due to some uniformly applied 

 constricting pressure from without. In all of these cases the points 

 of contact between the circles in the diagram will be extended into 



* Ricreatione delV occhio e delta mente, nelV Osservatione delle Chiocciole, Roma, 

 1681. 



