169 



inn ibcr of individuals do particular aggregates make their appearance. 

 Th ^e apparent associations of crystals are always the same and 

 quite characteristic for the crystals under consideration. Their 

 .iti.iu is governed by strict laws, and such "compound crystals" 

 show a constancy of their interfacial angles and mutual orientation 

 rith a degree of perfection analogous to that which is met with 

 in the properties of so-called single individuals. 



Such apparent aggroupments consist always of two or more 

 individuals of the same kind 1 ), every two of which are symmetrically 

 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 a twin 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 

 longest studied, are just the rather simple, real twins. In quite early 

 times a distinction was introduced 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 individuals 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 twinning-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 muscovitr, orpiment, stncblende, 

 phlogofnte, brucite, talc, leadhillite, sylvine, rock-salt and pyrophyllite, if brought 

 into immediate contact with them. 



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

 Molekularphysik /. p. 293-407. (1888). 



