represents a family of parallel planes. If the crystal possess symmetry with respect to the plane of two of the axes, (say the b-axis and the c-axis), then the existence of a plane face, forming one member of the family (h, k, l) requires the existence of a face forming one member of the family (- h, k, l). The collection of all the planes required by this law forms a complete or holohedral crystal form. Of equal importance are the partial crystal forms arrived at by the selection of certain planes from those of any complete crystal form. If half the planes be selected the resulting form is said to be hemihedral, if one quarter tetartohedral; the half or quarter selected must however be chosen according to certain rules, depending on the symmetry of the crystal. If, when the axes are suitably chosen, any one of the axial ratios become rational, it is clear that this ratio may be taken to be unity, and the two axes concerned are said to be equivalent; if further these axes be normal to planes of symmetry, they are said to be equivalent axes of symmetry. The law of selection of planes to make a hemihedral form is that only such planes can occur as intersect equivalent axes of symmetry at the same distance from the origin, at the same inclination, and in equal numbers'. The selection of half the planes of a complete crystal form may either include or exclude pairs of parallel planes; in the former case the resulting form is said to be hemihedral with parallel faces, in the latter hemihedral with inclined faces.

In the theory of elastic crystals, it is convenient to introduce two sets of rectangular axes. The axes of (x, y, z) are perfectly general, and the axes of (x, y, z。) are parallel to lines to which it is convenient to refer the faces of the crystal, (sometimes, but not always, crystallographic axes). We shall denote the displacements, stresses, and strains, referred to the latter system, by (uo, vo, Wo), (Po, Q., R., S。, To, U.), and (eo, fo, go, do, bo, Co); and the most general system of elastic constants corresponding to (16), when referred to the axes of (xo, Yo, zo), will be denoted by a's with double suffixes instead of c's.

1 An example will make this clearer. If no two axes of symmetry be equivalent, but three planes of symmetry be present, as in the rhombic system, a complete form is the octahedron (±1, ±1, ±1). A possible method of hemihedrism is by selection of the planes (±1, ±1, +1). If there be three equivalent axes of symmetry at right angles, as in the cubic system, this is not a possible method of hemihedrism.

Complete crystal forms are divided into six systems according to their symmetries. We shall exhibit the stress-strain relations for each of these systems, and for the most important related hemihedral forms, with reference to specially selected sets of axes of (xo, yo, 20).

To do this we shall follow F. E. Neumann in his assumptions that crystallographic symmetry is identical with symmetry in elastic quality, and the directions of equivalent axes of symmetry are elastically interchangeable.

33. Triclinic, Anorthic, or Doubly-Oblique System(21 Constants).

In this system there is no plane of symmetry, and no reduction takes place in the number of constants. The relations (16) with Crs Car are the stress-strain relations.

=

34. Monoclinic or Oblique System-(13 Constants).

0

This system possesses one plane of symmetry. Let this be the plane (xo, yo), then Po, Q., R。, U, must remain unaltered, and the other stresses must change sign, when z, and w, are changed into — z。 and — w。, i.e. when a and b, are changed into — a。 and — b。. Hence the coefficients α14, α15, 24, A25, A34, A35, A64, α vanish, and the energy-function W is given by

2W=(A11, A22, A33, A66, A12...Xeo,fo, go, Co)2 + (A4, A55, A45)αo, bo)2 (20), i.e. 2 W consists of a complete quadratic function of e, fo, go, Co, and a complete quadratic function of 。, bo.

Let (h, k, l) denote any plane of a complete form of this system, referred to the axes of (x。, Yo, z。), then (h, k, − 1) must be a plane of the form, and, taking the two parallel planes (―h, k, l) and (— h, — k, l), we get the complete crystal form. These planes do not form the boundary of a crystal since they do not enclose a space. The faces of a crystal are generally the sets of planes belonging to several complete or partial forms of the same system.

The parallel-faced hemihedral forms would consist of the planes (h, k, l) and (— h, — k, — l), or of the other pair. Each of these is identical with a complete triclinic form, and may therefore be rejected from our enumeration. If there were true monoclinic crystals exhibiting this mode of hemihedrism we could have the phenomenon of the combination of an apparently monoclinic form with an apparently triclinic, which has never been observed. We

shall in like manner reject all partial forms arrived at geometrically, which are identical with forms belonging to a different system of crystals.

Fig. 5.

The inclined-faced hemihedral forms would consist of the planes (h, k, l) and (—h, — k, l) or of the other pair, or again of the pair (h, k, l,) and (h, k, l) or of the other pair. The first named have no plane of symmetry.

Some inclined-faced hemihedral forms of this system possess no plane of symmetry, but the figure of any such form will be similarly situated with respect to the axes if it be rotated through two right angles about the z axis. It follows that Po, Q., R。, U。 remain unaltered, and S., T, change sign when uo, vo, o, yo are changed into -Uo, — Vo, — Xo, yo while w。 and z remain unaltered, i.e. when a and b are changed to a and b。. Hence the stress-strain formulæ for these are the same as for the complete forms of the same system. The remaining inclined-faced hemihedral forms of

this system possess one plane of symmetry, so that the formula (20) holds for all forms of this system.

35. Rhombic or Prismatic System-(9 Constants).

The complete forms of this system possess three planes of symmetry at right angles to each other. Let the planes (x, y) and (x, z) be planes of symmetry. Then all the coefficients a14, 15, 16, 24, A25, A26, A34, Aз5, A36, A45, 46, a56 vanish, and the energy-function is given by

2 W= (α11, A22, A33, A23, αз1, α12Xeo, fo: 90)2+ɑ44αo2+ɑ55bo2 + ɑ66 Co2 (21), which is the same as when there are three planes of symmetry. Topaz and Barytes are examples of crystals for which formula (21) holds.

Let (h, k, l) denote any plane of a complete form of this system referred to the axes (xo, Yo, zo), then (±h, k, l) must all be

C

Α'

A

Fig. 6

planes of the form, and the complete crystal form is the octahedron with rectangular diagonals of different lengths.

The types of the possible hemihedral forms are:

1o. The tetrahedron formed by the planes ABC, A’BC', A'B'C, AB'C', where AA', BB' and CC' are the axes. This has no plane of symmetry but the figure will come into a similar position with respect to the axes after a rotation through two right angles about either axis, hence for these inclined-faced hemihedral forms formula (21) holds.

2o. The half-form whose planes are ABC, ABC', A'B'C, A'B'C'. This is identical with the complete monoclinic form, and may therefore be rejected.

3°. The half-form whose planes are ABC, A'BC, A'B'C, AB'C. This has two planes of symmetry, and formula (21) holds.

We shall write (21) in the form

2W = (A, B, C, F, G, HXe。, ƒ。, 9.)2 + La2 + Mb2+Nc2.......(21). In this notation Cauchy's relations are

L=F, M=G, N=H.

36. Quadratic or Tetragonal System-(6 Constants). This system has three rectangular planes of symmetry, and two of the axes are equivalent; let these be the axes of x and yo, then P. must be the same function of e, that Q, is of fo, and P, and Q. must have the same term in g。; also S, must be the same function of a, that T, is of bo; we thus get the equalities anα22, α13 A23, α1 = α55, and W is given by the equation

0

=

=

2 W = (α11, α11, Asз, A2, A23, α12Xеo, fo, go)2 +ɑss (αo2+b2)+ቤ2.......(22).

Let (h, k, l) denote any plane of the complete form. Then the complete form will also contain the planes (±h, ±k, ±l) and (±k, ±h, ±l). If k=h the figure is an octahedron with rectangular diagonals two of which are equal in length. The hemihedral forms derivable from the most general complete form are obtained as follows:

1o. Selection by alternate planes. Of the 8 bounding planes, 4 are obtained from the form (h, k, l) by taking the signs all positive, or two negative and one positive, and the other 4 are obtained from these by interchanging h and k and at the same time