438 Proceedings of the Royal Society of Edinburgh. [Sess. 
xi. The following lines cannot be twin axes : — 
2 Uc (in II Uc, III Be), 4 Uc (in IV Uc), 6Uc (in VI Uc), 2Bc (in II Be, 
IV Be, VI Be, C Uc, G Be), 4 Be (in IV Be, G Be), 6Bc (in VI Be), because 
reversals relatively to the line, to the plane to which it is normal, and to 
a point are all co-directional operations of the structure. For instance, the 
normal to the clinopinakoid in orthoclase and the normal to the rhombic 
dodecahedron faces in spinel cannot be twin axes. 
§ 11. General Relation between Equivalent Twinning Operations. 
i. If the same relation between the component structures of a twin 
crystal results from different twinning operations, these are said to be 
equivalent to one another. 
ii. As all twinning operations are reversals (or equivalent to reversals), 
the combination of any two equivalent twinning operations must be a co- 
directional operation of the structure (§ 5, ii.); and if a co-directional 
operation be resolved into two reversals, either of these acting alone on the 
structure will be equivalent to the other, so that if one be a twinning opera- 
tion, the other will be an equivalent twinning operation, and in like manner, 
if one be a co-directional operation, the other will be so likewise. If 
therefore, every combination of two reversals (see § 5) be examined to 
discover if it be a co-directional operation, every case in which two 
reversals are equivalent twinning operations will be ascertained. 
§ 12. Equivalent Twinning Operations deduced from Group (a) 
of Three Reversals (§5, iii.). 
i. The operation of reversal relatively to a point is equal to reversal 
relatively to a plane combined with reversal relatively to its normal. 
If, therefore, a twin crystal result from the reversal relatively to a plane 
of a portion of a structure possessing point symmetry , an indistinguishable 
result may be obtained by substituting a reversal relatively to the normal 
to the plane, and vice versa. The same twin plane and axis will correspond 
to both operations (§ 10^ ii.). Inversely, if there be both plane and line 
twinning with the same twin plane and twin axis, the untwinned structure 
must possess point symmetry (see, for example, figures lUc and 1 UcPL, 
p. 435). It follows from this that point twinning cannot coexist with plane 
and line twinning which have a common twin axis (§ 10, iv.). 
ii. A reversal relatively to a line is equal to a reversal relatively to the 
plane to which it is normal combined with reversal relatively to a point. 
