456 Proceedings of the Royal Society of Edinburgh. [Sess. 
systems ; thus prolectite, chondrodite, and clinohumite have the symmetry 
of the monoclinic system, but the three axes are at right angles to one 
another ; a reversal, therefore, relatively to the vertical axis would be a 
co-spatial operation. Every such co-spatial operation will be a reversal 
and therefore a twinning operation ; except in one possible case in which 
it would be a rotation with cyclic number 4. 
§ 28. Combinations of Co-spatial and Twinning Operations. 
We may now consider the case of co-spatial planes formed by a co- 
spatial operation followed by an operation which would have converted 
a plane into a twin plane if the co-spatial operation had not preceded it. 
If these two operations are reversals related to one another in the manner 
described in § 19, their combination will be equal to another reversal, and 
the compound crystal containing the resulting co-spatial plane will be 
a twin crystal. If, on the other hand, they are reversals which are not 
so related, the operation giving rise to the co-spatial plane will be such 
a rotation as is described in § 20. 
Example . — If a portion of a quartz crystal be reversed relatively to the vertical 
axis, this will be a co-spatial operation and also a twinning operation (see figures 3 Uh 
and 3 UhL, p. 434). If the same portion be then reversed relatively to the normal 
to a rhombohedron face, these two operations will together be equivalent to a 
rotation round a line at right angles to the vertical axis and to the normal to the 
rhombohedron face, that is to say, round the normal to a face of a trigonal prism of the 
second order, through an angle twice that between the normal to the rhombohedron 
face and the vertical axis. The rhombohedral face in question will be a co-spatial 
plane. 
Such a combination is believed by Professor v, Goldschmidt to occur in quartz 
twins with oblique twin axes ( Tsch . Min. Pet. Mitt., vol. xxiv., 1905, pp. 157-182), 
but might be explained as the result of the combination of two independent 
twinning operations. 
§ 29. Cross-spatial Planes. 
A cross-spatial plane may be formed as the result of a co-spatial 
operation followed by an operation which would have given rise to a 
cross plane if the co-spatial operation had not been already applied. As a 
co-spatial operation is usually a twinning operation, and an operation 
resulting in a cross plane is always one, the combined operation will be 
as a rule a twinning operation or rotation according as the combination 
falls under § 19 or § 20. 
