316 THE FORMS OF TISSUES [ch. 



and angles presented by solid figures in symmetrical juxtaposition. 

 Let us take a simple case of the latter kind, and again afterwards, 

 so far as possible, let us try to keep the two themes separate. 



Where we have three spheres in contact, as in Fig. 114 or in 

 either half of Fig. 116, B, let us consider the point of contact 

 (0, Fig. 114) not as a point in the plane section of the diagram, but 

 as a point where three furrows meet on the surface of the system. 

 At this point, three cells meet ; but it is also obvious that there meet 

 here six surfaces, namely the outer, spherical walls of the three 

 bubbles, and the three partition-walls which divide them, two and 

 two. Also,/oMr lines or edges meet here ; viz. the three external arcs 

 which form the outer boundaries of the partition- walls (and which 

 correspond to what we commonly call the "furrows" in the seg- 

 menting egg) ; and as a fourth edge, the "arris" or junction of the 

 three partitions (perpendicular to the plane of the paper), where 

 they all three meet together, as we have seen, at equal angles of 

 120°. Lastly, there meet at the point four solid angles, each 

 bounded by three surfaces : to wit, within each bubble a solid 

 angle bounded by two partition- walls and by the surface wall ; 

 and (fourthly) an external solid angle bounded by the outer 

 surfaces of all three bubbles. Now in the case of the soap-bubbles 

 (whose surfaces are all in contact with air, both outside and in), 

 the six films meeting at the point, whether surface films or partition 

 films, are all similar, with similar tensions. In other words the 

 tensions, or forces, acting at the point are all similar and symmet- 

 rically arranged, and it at once follows from this that the angles, 

 solid as well as plane, are all equal. It is also obvious that, as 

 regards the point of contact, the system will still be symmetrical, 

 and its symmetry will be quite unchanged, if we add a fourth 

 bubble in contact with the other three : that is to say, if where 

 we had merely the outer air before, we now replace it by the air 

 in the interior of another bubble. The only difference will be that 

 the pressure exercised by the walls of this fourth bubble will alter 

 the curvature of the surfaces of the others, so far as it encloses 

 them ; and, if all four bubbles be identical in size, these surfaces 

 which formerly we called external and which have now come to 

 be internal partitions, will, like the others, be flattened by equal 

 and opposite pressure, into planes. We are now dealing, in short. 



