DS, CRYSTALLOGRAPHY. 
cube and other forms with reference to these axes, the following facts 
will become apparent. 
in the cube (fig: 1) the front plane touches the extremity of axis a, 
but is parallel to axes 6 and c. When one line or plane is parallel to 
another they do not meet except at an infinite distance, and hence the 
sign for infinity is used to express parallelism. Employing 7, the 
initial of infinity, as this sign, and writing c, >, a, for the semi-axes so 
lettere1, then the position of this plane of the cube is indicated by the 
expression 7¢: 2b: la. The top and side-planes of the cube meet one 
axis and are parallel to the other two, and the same expression answers 
for each, if only the letters a, 6, c, be changed to correspond with their 
positions. The opposite planes have the same expressions, except that 
the c, b, a will refer to the opposite halves of the axes and be -¢, -0, -a. 
In the dodecahedron, fig. 15, the right of the two vertical front planes 
t, meets two axes, the axes @ and 0, at their extremities, and is parallel 
to the axis¢. Hence the position of this plane is expressed by t¢: 10: la. 
So, all the planes meet two axes similarly and are parallel to the third. 
The expression answers as well for the planes? in figs. 18, 14, as for that 
of the dodecahedron, for the planes have all the same relation to the axes. 
In the octahedron, fig. 11, the face 1, situated to the right above, 
like all the rest, meets the axes @, 0, ¢, at their extremities; so that the 
expression 1c: 10: 1a answers for all. 
Again, in fig. 17 (p. 20) there are three planes, 2-2, placed symmet- 
rically on each angle of a cube, and, as has been illustrated, these are 
the planes of the trapezohedron, fig. 19. The upper one of the planes 
2-2 in these figures, when extended to meet the axes (as in fig. 19), 
intersects the vertical ¢ at its extremity, and the others, @ and 0, at 
twice their lengths from the centre. Hence the expression for the plane 
is le: 2b: 2a. So, as will be found, the left hand plane 2-2 on fig. 
17, will have the expression 2¢: 10: 2a; and the right hand one, 
2c: 2b: 1¢. Further, the same ratio, by a change of the letters for the 
semi-axes, will answer for all the planes of the trapezohedron. 
In fig. 20 there are other three planes, 2,.0n each of the angies of a 
cube, and these are the planes of the trisoctahedron in fig. 21. The 
lower one of the three on the upper front solid angle, would meet if 
extended, the extremities of the axes a and 6, while it would meet the 
vertical axis at twice its length from the centre. The expression 
2c: 1b: 1a@ indicates, therefore, the position of the plane. So also, 
le: 1b: 2a and 1c: 25: 1@ represent the positions of the other two 
planes adjoining; and corresponding expressions may be similarly ob- 
tained for all the planes of the trisoctahedron. 
Again, in fig. 39, of the cube with two planes on each edge, and in 
fig. 31, of the fetrahexahedron bounded by these same planes, the left 
of the two planes in the front vertical edge of fig. 30 (or the corre- 
sponding plane on fig. 31) is parallel to the vertical axis ; its intersections 
with the lateral axes, @ and 4, are at unequal distances from the centre, 
expressed by the ratio 2): 1a. This ratio for the plane adjoining on 
the right is 1): 2a. The position of the former is expressed by the 
ratio 7¢ : 20: 1a, and for the other by 7c: 10: 2a. Thus, for each of 
the planes of this tetrahexahedron the ratio between two axes is 1 : 2, 
while the plane = eee to the third axis. 
Again, in fig. 22, of the cube with six planes on each solid angle, and 
in the hexoctahedron in fig. 23, made up of such planes, each of the 
planes when extended so that it will meet one axis at once its length 
