changing the sign of l. The resulting inclined-faced hemihedral form has no plane of symmetry, but by rotating it round the z。 axis through 90° it will come into a similar position with respect to the axes. We shall see presently that equation (22) does not hold for this case (fig. 7).

In fig. 7 AB and A'D are two sides of a rhombus, two of whose corners are A, A', and the other two are on OC equidistant from 0. BC and CD are two sides of an equal rhombus, one of whose corners is C, another on OC opposite to C, and the other two on AA'. The complete form consists of two pyramids, vertices Z and Z standing on the 8 sided figure ABCDA'... The hemihedral forms obtained by the first method of selection are such as that bounded by the planes Z'AB, ZBC, Z'CD, ZDA'...

2o. Selection by alternate pairs intersecting in the principal plane of symmetry (x, y). Of the 8 bounding planes, 4 are obtained from (h, k, l) by taking the first two letters either (h, k) or (h, k) or (k, h) or (-k, h), and the other 4 from these by changing the sign of l. The resulting parallel-faced hemihedral form has three planes of symmetry, viz.: the plane (x。, y。) and

planes parallel to the sides of the square base in this plane, and formula (22) holds for these forms (fig. 8).

3°. Selection by alternate octants. Of the 8 bounding planes, 4 are obtained by taking the signs of (h, k, l) all positive or two negative and one positive, and the other 4 are obtained from these by interchanging h and k. The resulting inclined-faced hemihedral forms have two planes of symmetry which bisect the angles between the planes (x, z) and (yo, zo), but the equivalence of the corresponding axes of symmetry is lost. The form therefore belongs really to the rhombic system and may be rejected here.

For the trapezohedral hemihedrons obtained by the first method of selection it can be shewn that the energy-function is given by an equation of the form

2 W=(α11, α11, αsз, A23, A23, α12Xeo,ƒfo, go)2+Ass(αo2+b2)+α66Co2+2α16(Co−ƒo) Co

.(23),

which involves 7 constants. This mode of crystallization has been observed in certain organic salts only.

37. Regular or Cubic System-(3 Constants).

This system has three rectangular planes of symmetry and all three axes equivalent. It is clear that W is given by the equation

2W = (α11, α, α11, α12, α12, α12 Xeo, ƒo, 9o)2 +ɑss (αo2 +b2+c2).....(24). Let (h, k, l) denote any plane of the crystal. Then the complete form is obtained by taking the six permutations of the letters h, k, l, and giving either sign to each letter. The most general complete form is therefore bounded by 48 planes. The types of hemihedrism are similar to those of the tetragonal system. If the first method (by alternate planes) be adopted, the resulting figure will have no plane of symmetry; but it will coincide with its original position after a rotation through 90° about either axis, and equation (24) holds for this case. If the second method (by alternate pairs intersecting in a principal plane of symmetry) be adopted, the resulting parallel-faced hemihedral forms will have three rectangular planes of symmetry, one belonging to the complete form, and the other two bisecting the angles between two principal planes of the complete form, and all three axes equivalent, and equation (24) will clearly hold for this case. If the third method (by alternate octants) be adopted the resulting inclined-faced hemihedral forms will be such that, by a rotation. through 45° about either axis, the two principal planes of the complete form, that meet in that axis, become planes of symmetry, and equation (24) will therefore hold for this case.

Fluor-spar, Rock-salt, Pyrites, and Potassium Chloride are examples of minerals for which formula (24) holds.

38. Hexagonal System-(5 Constants).

This system has 7 planes of symmetry, of which one is perpendicular to the axis z。, and 6 meet in the axis z, and are symmetrically arranged round that axis, and the axes perpendicular to the latter 6 planes are equivalent. We can express this by beginning with the rhombic system, and supposing that the expressions for the stresses in terms of the strains are unaltered by a rotation

the equations of transformation of strain-components, given in (33)

g' = 9,

The equations of transformation of stress-components given in (14) of art. 16, give P', Q',... in terms of P, Q,... If we write down the corresponding formulæ for P, Q,... in terms of P', Q',... we shall get

P = † P' + & Q' — § √3 U',

Q = &P' + {Q' + ±√3U',

R = R',

S = †S' + }√3T",

Now writing equation (21) in the form

2 W = (A, B, C, F, G, Heo, fo, 9%)2 + La2+Mb2+ Nc2,

(27).

substituting for S' and T' in the S, T equations of (27), and equating coefficients of a or b, we obtain L = M.

Substituting for R' in the R equation of (27), and equating coefficients of e or ƒ, we obtain F= G.

Substituting for P', Q', U' in the P, Q equations of (27), and equating coefficients of e in the P equation, and coefficients off in the equation, we get 3N1AB-H, and ‡N=1B-2A-§H, from which_A =B, and N= } (A − H).

Thus the energy-function is given by

=

16

Note that this formula is unaltered by turning the axes of x, and yo through any angle.

Beryl is an example of a crystal for which this formula holds.

39. Rhombohedral System-(6 Constants).

The most important hemihedral forms of the hexagonal system are the rhombohedrons obtained from a hexagonal pyramid by

the selection of alternate planes. In figure 9 ABCA'B'C' is a regular hexagon, and ZZ a perpendicular axis, and the faces of the rhombohedron are ZAB, Z'BC, ZCA', Z'A'B', ZB'C', and Z'C'A. These forms are unaltered by rotation through 120° about the axis zo, and also by rotation through 180° about the line AA', which we take for the axis . We have already seen (art. 34) that the x。. last property produces just the same simplification in the energyfunction as if the plane x=0 were a plane of symmetry, and we may therefore set out from the form

2 W = (α11, α22, αзз, ass ...Xeo, ƒo, go, αo)2+(A55, α 66, A56(bo, Co)2. If we work out the conditions that this may be unaltered by a