446 Proceedings of the .Royal Society of Edinburgh. [Sess. 
the plane containing the twin axes, the line at right angles to them, and 
the angle between them remain the same. The successive application of 
the two twinning operations therefore results in a combination of T x and T\, 
where T\ is a twinning operation with the same mode of twinning as T 2 , but 
with a 2 as twin axis. This combination will be equal to a rotation in the 
negative direction a 2 , a v round the same line, combined as before, if the 
modes of twinning be unlike, with a reversal relatively to a point. Then 
T v T' 2 = G -20 (or (7-20. Ri) = T 2 .T v If T 2 be applied first, a x will be moved 
in like manner to a\ on the other side of a 2 , and the rotation will be 
in the positive direction a v a 2 . Therefore T 2 . T\ = C 2 e (or G 2 e • Ri) = T l . T 2 , 
where T\ is a twinning operation similar to T x but with a\ as twin 
axis. 
ii. A twin crystal subjected to two independent twinning operations, 
the result of the successive application of which depends on the order of 
their application, may consist of five differently oriented portions— one, 
S, to which neither has been applied, two, S v S 2 , to which one, and two, 
$ 3 , $ 4 , to which both have been applied. The relations between these are 
shown in the following table : — 
Disposition of the structure to 1 
which the operations are applied i 
Resulting disposition of the 
structure 
S 
s, t y 
s 2 ^2 
£2 
T, T. 
1 
C 2 e 1 
C- 2 e r 2 t 1 
t 2 t\ 
C-29 \ 
2 w 2 9 0-2 9 
C-29 T\ 1\ 
1 
2 
T\ 
G — 40 
1 
Operations 
applied. 
If the two modes of twinning be different, the rotations 20 and —20 
will be combined with a reversal relatively to a point. The above relations 
are illustrated by the twinning of bournonite and marcasite on dome faces 
aragonite on prism faces ; and combinations of albite and pericline twinning 
in plagioclase. 
§ 21. Cross Planes in Twin Crystals. 
i. The plane of contact in twin crystals is not always a twin plane. 
It may be a plane in which some, but not all, of the lines are common 
lines along which the molecular intervals are the same in the two structures 
and determine their regular intergrowth. Even where the plane of contact 
is a twin plane, it is probably only the more important common lines 
(that is to say, those with the smallest molecular intervals) that determine 
the twinning. 
