280 
MR. MACQUORN RANKINE ON AXES OP 
If there be a pair of such additional axes in one of the planes of orthotatic sym- 
metry only, those axes are normal to the lateral faces of a Right Rhombic Prism. 
26. Of Cybo'id Symmetry. 
The case of Cybo'id Symmetry is that in which the Homotatic Coefficients are equal 
for three Orthogonal Axes, viz. — - 
M = (/3’) = (r’‘) ! (/3r)+2(X’) = (y«=)+2(,».^) = («^)H-2(.’) ; 
2 {^v) -j- (aX) “ 2 (vX) -j- ~ 2 (Xf) -f- {<yv) =0 (48.) 
In this case, the Principal Metatatic Axes coincide with the Principal Euthytatic 
Axes, which are normal to the faces of a cube j the Diagonal Metatatic Axes, normal 
to the faces of a regular Rhombic Dodecahedron, are Euthytatic also ; and there are, 
besides, four additional euthytatic axes symmetrically situated between the first nine, 
and normal to the faces of a regular octahedron, making in all thirteen euthytatic 
axes. 
27 . Of Monaxal Isotropy. 
Monaxal Isotropy denotes the case in which the homotatic coefficients are com- 
pletely isotropic round one axis only. In this case, the principal euthytatic axes are, 
the axis of isotropy, and every direction perpendicular to it ; and when there are 
additional axes, determined as in the preceding articles, they are normal to the sur- 
face of a cone. 
28. Of Complete Isotropy. 
In the case of Complete Isotropy of the Homotatic coefficients, every direction 
is a euthytatic axis. 
29. Probable Relations between Euthytatic Axes and Crystalline Forms. 
In the preceding articles it has been shown, what must be the nature of the rela- 
tions between the fifteen homotatic coefficients, for various solids, having systems of 
euthytatic axes normal to the faces and edges of the several Primitive Forms known 
in Crystallography. 
It is probable that the normals to Planes of Cleavage are Euthytatic Axes of Mini- 
mum Elasticity. 
It may also be considered probable, that in some cases, especially in the Tessular 
System, which corresponds to Cybo'id Symmetry, and in the case of the pyramidal 
summits of crystals of the Rhombohedral System, Euthytatic Axes correspond to 
symmetrical summits of crystalline forms. In the icositetrahedral crystals of leucite 
and analcime, and the tetracontaoctahedral crystals of diamond, there are twenty-six 
symmetrical summits, one pair corresponding to each of the thirteen axes of cybo'id 
symmetry. 
