44 7 
1911-12.] The Geometry of Twin Crystals. 
ii. Let S', S" be two structures of the same shape which form a 
compound crystal containing a common plane de made up of two equivalent 
planes d'e and d"e" of S' and S" respectively, and let d be a common line in 
de composed of the two equivalent lines d' and d" (see figures, p. 448). Then 
the cyclic succession of the lines in d'e' on either side of the line d' will 
be either the same as that of the lines in d"e" on either side of d" , or 
opposite. In the former case, since d' coincides with d", every other line 
in d'e must coincide with an equivalent line in d"e". Every line in the 
common plane de is therefore a common line and the common plane is 
co-linear, so that, if the two structures are not co-directional, it will be 
a twin plane. 
iii. If, on the other hand, the cyclic succession of the lines of the plane 
d'e' on either side of d' be opposed to that of the lines in the plane d"e" on 
either side of d", the lines of d'e will not all coincide with equivalent lines 
of d"e r ' and the common plane de will not be co-linear. If, however, the 
operation of reversal relatively to the line d' be applied to the structure S', 
the cyclic succession of the lines in the plane d'e' will be reversed, so that 
the cyclic succession will become the same on either side of d' in d'e' and d" 
in d"e", and as d still remains a common line, every line in d'e' will coincide 
with an equivalent line in d"e" and form a common line, and the common 
plane de will be co-linear. Every common plane which is not co-linear 
but in which there is at least one common line may therefore be converted 
into a co-linear common plane by the reversal of one of the component 
structures relatively to the common line. 
iv. If the structure S' be again reversed relatively to the same line, the 
original common plane will be obtained. In this second reversal not only 
the line d but the line e at right angles to it in de will remain a common 
line. If, therefore, there be one common line in a common plane, there 
must be another at right angles to it. A common plane which is not 
co-linear but contains one or more pairs of common lines may be termed 
a cross 'plane . In figures 1 UuP' d , 1 U uL' d , 1 UuP' e , and 1 UuL' e , the 
short lines d and e intersecting at the centre are common lines and 
the plane of projection is in each case a cross plane. The discs with 
hollow centres indicate the original disposition of the structure as shown 
in figure 1 Uu, p. 432, while those with solid centres show the disposition 
of a portion of the structure to which the operation of plane or line 
twinning with twin axis d or e, as the case may be, has been applied ; 
see § 22. 
v. Every plane containing one or more pairs of common lines at right 
angles to each other will be a common plane, and if it is not co-linear 
