714 ON CONCRETIONS, SPICULES, ETC. [ch. 



AF will be edges of the spherical tetrahedron, and will be the 

 centre of s)niimetry. Make the angle FOG = FOA = EOA. Cut out 

 the circle EAFG, and cut through the radius E0\ fold at AO, FO, 

 GO, and fasten together, using EOG for a flap. Make four such 

 sheets, and fasten together back to back. The model will be much 

 improved if little cusps be left at the corners in the cuttmg out. 



The geometry of the little inner tetrahedron is not less simple 

 and elegant. Its six edges and four faces are all equal. The films 

 attaching it to the outer skeleton are all planes. Its faces are 

 spherical, and each has its centre in the opposite corner. The 

 edges are circular arcs, with cosine ^ ; each''is in a plane perpendicular 

 to the chord of the arc opposite, and each has its centre in the middle 

 of that chord. Along each edge the two intersecting spheres meet 

 each other at an angle of 120°*. 



This completes the elelmentary geometry of the figure; but one 

 or two points Temain to be considered. 



We may notice that the outer edges of the little skeleton are 

 thickened or intensified, and these thickened edges often remain 

 whole or strong while the rest of the surfaces shew signs of imper- 

 fection or of breaking away; moreover, the four corners of the 

 tetrahedron are not re-entrant (as in a group of bubbles) but a 

 surplus of material forms a little point or vcusp at each corner. 

 In all this there is nothing anomalous, and nothing new. For we 

 have already seen that it is at the margins or edges, and a fortiori 

 at the corners, that the surface-energy reaches its maximum — with 

 the double effect of accumulating protoplasmic material in the form 

 of a Gibbs's ring or bourrelet, and of intensifying along the same 

 Hues the adsorptive secretion of skeletal matter. In some other 

 tetrahedral systems analogous to Callimitra, the whole of the 

 skeletal matter is concentrated along the boundary-edges, and none 

 left to spread over the boundary-planes or interfaces: just as among 

 our spherical Radiolaria it was at the boundary-edges of their many 

 cells or vesicles, and often there alone, that skeletal formation 

 occurred, and gave rise to the spherical skeleton and its mesh work 



* For proof, see Lamarle, op. cit. pp. 6-8. Lamarle shewed that the sphere 

 can be so divided in seven ways, but of these seven figures the tetrahedron alone 

 is stable. The other six are the cube and the regular dodecahedron; prisms, 

 triangular and pentagonal, with equilateral base and a certain ratio of base to 

 height; and two polyhedra constructed of pentagons and quadrilaterals. 



