IX] 



OF JOHANNES MtJLLER'S LAW 



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(or some of them) by curves, the plane facets by slightly curved 

 surfaces, or the regular by non-equilateral polygons*. 



In some cases, such as Haeckel's Phatnaspis cristata (Fig. 342), 

 we have an elHpsoidal body from which the spines emerge in the 

 order described, but which is not obviously divided into facets. 

 In Fig. 234 I have indicated the facets corresponding to the rays, 

 and dividing the surface in the usual symmetrical way. 



6F, 



eF. 



-6F, 



Fig. 341. Dorataspis cristata Hkl. A, viewed according to Muller's law: a, four 

 polar plates; h, four intermediate or "tropical" plates; c, four equatorial 

 plates. B, an alternative description: Fg, two polar and six equatorial 

 hexagonal plates; F5, two rows of six intermediate pentagonal plates. 



About any polyhedron (within or without) we may describe 

 another whose corners correspond to the sides, and whose sides to 

 the corners, of the original figure; or the one configuration may 

 be developed from the other by bevelling off, to a certain definite 

 extent, the corners of the original polyhedron. The two figures, 

 thus reciprocal to one another, form a "conjugate pair," and the 

 principle is known as the "principle of duahty" in polyhedral . 

 Of the regular sohds, cube and octahedron, dodecahedron and 



* Muller's interpretation was emended by Brandt, and what is known as Brandt's 

 Law, viz. that the symmetry consists of two polar rays and three whorls of six 

 each, coincides so far with the above description : save only that Brandt says plainly 

 that the intermediate whorls stand equidistant between the equator and the poles, 

 i.e. in latitude 45", which, though not very far wrong, is geometrically inaecurate. 

 But Brandt, if I understand him rightly, did not propose his "law" as a substitute 

 for Muller's, but rather as a second law, applicable to a few special cases. 



t First proved by Legendre, EUm. de Geometrie, vii. Prop. 25, 1794. 



