174 



apparent associations of crystals are always the same and quite 

 characteristic for the crystals under consideration. Their formation 

 is governed by strict laws, and such "compound crystals" show a 

 constancy of their interfacial angles and mutual orientation with 

 a degree of perfection analogous to that, which is met with in the 

 properties of so-called single individuals. 



Such apparent aggroupments seem to consist always of two or 

 more individuals of the same kind 1 ), every two of which are symme- 

 trically arranged with respect to a net-plane or to a molecular row 

 of the space-lattice. They are commonly distinguished as twins. 



In the case of real twins, these are called twins of the first order, 

 if the symmetry-element of the compound crystal be a row of par- 

 ticles; if it is a net-plane, the twins are said to be of the second order. 



Now the examples of these crystal-aggregates which have been 

 most studied, are the rather simple, real twins. In quite early times 

 of the development of crystallography a distinction was intro- 

 duced between twins which appear to be formed by juxtaposition 

 of the composing individuals, and those made by their mutual 

 penetration. In the first case the molecular system of both indivi- 

 duals is considered to be symmetrical with respect to the plane 

 of juxtaposition or composition-plane, - - it may coincide with 

 the proper twinning-plane or not (gypsum). This composition-plane 

 may be parallel to the t winning-axis, or to the twinning-plane, 

 or it may be perpendicular to one of them; but it must always be 

 a possible crystallographical plane of the crystal, or a plane perpen- 

 dicular to a crystallographically possible edge of it. The common 

 straight line of both individuals may be a possible edge of the 

 crystal, or the normal to a possible crystal-face. 



Of course, a symmetry-axis of even period or symmetry-planes 

 of the crystal can never have the functions of a twinning-axis or 

 of a twinning-plane ; for in that case not twins, but parallel growths 

 of two crystals would be produced. 



In the second case the molecular systems of the two individuals 

 are, as it were, soldered together: the two crystals are symmetrical 



take, as a rule, regularly oriented positions with respect to the crystallographical 

 directions of cleavage-lamellae prepared from muscovite, orpiment, zincblende, 

 phlogopite, brucite, talc, leadhillite, syhrine, rock-salt and pyrophyllite, if brought 

 into immediate contact with them. 



x ) A. Sadebeck, Angewandte Krystallographie, p. 244, (1873); O. Lehmann, 

 Molekularphysik, I, p. 293407, (1888). 



