448 Proceedings of the Royal Society of Edinburgh. [Sess. 
will be a cross plane. For let d and e be two common lines at right angles 
to each other, de the plane containing them, and n its normal ; then since 
d is a common line ne to which it is normal will be a common plane 
(§7, iii.). But since e is a common line in the common plane ne, n which 
is at right angles to e will also be a common line, and the plane de to 
which it is normal will be a common plane. 
vi. A cross plane may be either a contact, or an abstract common plane 
(§8, iii.), the latter having usually no structural significance. 
vii. It follows from paragraphs iii. and iv. that if two structures 
possess a co-linear common plane, a reversal of one relatively to a line in 
this co-linear plane will convert it into a cross plane, provided that the 
reversal is not a co-linear operation of the component planes. This is also 
the case with a reversal relatively to a plane at right angles to the co-linear 
plane. 
