445 
1911-12.] The Geometry of Twin Crystals. 
These relations are illustrated by the twin crystals of nepheline described 
by H. Baumhauer (Zeit. Kryst. Min., 1882, vol. vi. pp. 210-6, and 1891, 
yoI. xviii. pp. 611-8), where T x is an operation of plane twinning with the 
faces of the hexagonal prisms of the first and second order as twin planes 
and their normals as twin axes ; T 2 is an operation of point twinning 
equivalent to plane twinning with the basal plane as twin plane and the 
principal axis as twin axis ; and T s is an operation of line twinning with the 
same twin planes and axes as T 1 (see p. 444, figures 6 Uu, QUuP' = 1 UhP — T v 
6 UuPI — T 2 , 6 TJuL' = 1 UhL = T 3 ; the symbols P' and L' indicate the presence 
of axes of plane and line twinning respectively lying in the plane of projec- 
tion. The short lines intersecting in the centre are common lines ; see § 21). 
vii. The well-known law (see § 8, ii. ; and § 24, ii.) that in every 
twin crystal there is either a twin plane parallel to a possible face or a 
twin axis parallel to a possible edge, or both, is frequently inapplicable to 
the twinning relations between the two portions of a twin crystal which 
are connected by a twinning operation which is a combination of two 
independent twinning operations. 
For instance, in crystals of plagioclase showing both albite twinning, and 
karlsbad twinning (with the vertical axis as twin axis), the twin axes of the two 
twinning operations are at right angles to each other. Their combination is there- 
fore a twinning operation with a twin axis lying in the hrachy-pinakoid at right 
angles to the vertical axis and neither normal to a possible face nor parallel to a 
possible edge. 
§ 20. Combinations of Independent Twinning Operations with 
Twin Axes Oblique to one Another. 
i. We will now consider the case of two independent twinning opera- 
tions such that no twin axis resulting from one is parallel or at right 
angles to a twin axis resulting from the other. 
Let T 1 and T 2 be the twinning operations, and a x and a 2 their respective 
twin axes making an angle 0 with each other. Each operation will be 
a reversal relatively to a plane or a line, and not a reversal relatively to 
a point or equivalent to one (§ 19, i.). If a x and a 2 both remained fixed, 
and the operation T x were first applied and then T 2 , the result would be 
a rotation through an angle 20 round a line at right angles to cq and a 2 in 
the direction a v a 2 , combined, if the modes of twinning were different 
with a reversal relatively to a point (§5, viii.). Thus T 1 .T 2 = C 2e (or C 2e .Ri), 
where C 2e is a rotation in the direction a v a 2 , which is taken as positive. 
The first twinning operation T x will, however, change the position of the twin 
axis a 2 of the second operation T 2 to a' 2 on the other side of a v but so that 
