22 CRYSTALLOGRAPHY. 



cube and ether forms with reference to these axes, tho following facta 



will become apparent. 



In the cube (fig. 1) the front plane touches the extremity of axis «, 

 but is parallel to axes b 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 i, the 

 initial of infinity, as this sign, and writing c, b, a, for the semi-axes so 

 lettered, then the position of this plane of the cube is indicated by the 

 expression ic ; ib : la. The top and side -planes of the cube meet one 

 axis and are parallel to the other two, and the same expression answera 

 for each, if only the letters a, b, c, be changed to correspond with their 

 positions. The opposite planes have the same expressions, except that 

 the c, 5, a will refer to the opposite halves of the axes and be -c, -5, -«. 



In the dodecahedron, fig. 15, the right of the two vertical front planea 

 ft. meets two axes, the axes a and b, at their extremities, and is parallel 

 to the axis c. Hence the position of this plane is expressed by ic : 1 b : la. 

 So, all the planes meet two axes similarly and are parallel to the third. 

 The expression answers as well for the planes i in figs. 13, 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 a, b, c, at their extremities ; so that the 

 expression Ic : lb : la 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 c at its extremity, and the others, a and b, at 

 twice their lengths from the centre. Hence the expression for the plane 

 is lc : 2b : 2a. So, as will be found, the left hand plane 2-2 on fig. 

 17, will have the expression 2c : lb : 2a ; and the right hand one, 

 2c : 2b : la. 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, on each of the angles 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 &, while it would meet the 

 vertical axis at ticice its length from the centre. The expression 

 2c : lb : la indicates, therefore, the position of the plane. So also, 

 lc : lb : 2a and lc : 2b : la 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. 30, of the cube with two planes on each edge, and in 

 fig. 31, of the tetrahexahsdron 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, a and b, are at unequal distances from the centre, 

 expressed by the ratio 2b : la. This ratio for the plane adjoining on 

 the right is lb : 2a. The position of the former is expressed by the 

 ratio ic.2b: la, and for the other by ic : lb : 2a. Thus, for each of 

 the planes of this tetrahexahedron the ratio between two axes is 1 : 2, 

 while the plane is parallel 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 plar.es, each of the 

 planes when extended so that it will meet one axis at once its length 



