ELASTICITY AND CRYSTALLINE FORMS. 
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and that the value of u which makes it vanish shall neither be 0° nor 90° ; that is to 
say, that the second member of the above equation (46.) shall lie between +1 and 
— 1 ; in which case the equation is satisfied by equal values of a with opposite signs. 
Hence are deduced the following theorems, which are stated in such a form as to be 
applicable to planes of symmetry, whether hexagonal or otherwise. 
If, in any -plane of tasinomic symmetry containing a pair of Orthogonal Euthy- 
tatic Axes, the difference of the Euthytatic coefficients for these axes be equal to or 
greater than the Metatatic Difference, there are no additional euthytatic axes in that 
plane. 
If, on the other hand, the difference of such Euthytatic coefficients be less than the 
metatatic difference, there are, in such plane of symmetry, a pair of additional euthy- 
tatic axes making with each other a pair of angles bisected by the orthogonal euthytatic 
axes. 
2a is the angle bisected by the axis Oy^. 
In the case of Hexagonal Symmetry, the additional axes thus found are normal to 
the faces of one pyramidal dodecahedron, and the edges of another. 
24. Of Orthorhombic Symmetry. 
Let a solid have one of the three principal euthytatic axes, O^i, normal to the 
other two, Ox^, Opi ; let the last two be oblique to each other, and have equal sets 
of homotatic coefficients, viz. — 
(/3y).+2(X*), = (y«),+2(f.^).i 2(rt + («^)=2(.X) + (fr), (47.) 
then that solid may be said to have Orthorhombic Symmetry, its principal euthytatic 
axes being normal to the faces of a right rhombic prism. 
The existence or non-existence, and the position, of a pair of additional euthytatic 
axes in the longitudinal planes of yiZ^, z^x^, is to be determined as in the preceding 
article. When such axes exist, they are normal to the faces of an Octahedron with a 
Rhombic Base. 
25. Of Orthogonal Symmetry. 
If the three principal Euthytatic Axes be orthogonal, they are normal to the faces 
of a right rectangular or square prism, and to the edges of a right rhombic or square 
prism. The existence or non-existence, and position, of a pair of additional euthy- 
tatic axes in each of the principal planes of such a solid, are determined as in 
article (23.). 
If there be a pair of such additional axes in each of the three principal planes, they 
are normal to the faces of an irregular Rhombic Dodecahedron, and to the edges of a 
Rhombic Octahedron. 
If there be a pair of such additional axes in two of the three principal planes, those 
axes are normal to the faces of an Octahedron with a Rectangular or square base, and 
to the edges of an Octahedron with a Rhombic or square base. 
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