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DIMETRIC, OR TETRAGONAL SYSTEM. 33 
2. Positions of the Planes with reference to the Axes,— Let- 
tering of planes. In the prism fig. 10, the lateral planes are parallel to 
the vertical axis and to one iateral axis, and meet the other lateral axis 
at its extremity. The expression for it is hence (c¢ standing for the 
vertical axis and a, } for the lateral) ic : ib : 1a, 7, as before, standing 
for infinity and indicating parallelism. For the prism of fig. 12, the 
prismatic planes meet the two lateral axes at their extremities, and 
are parallel to the vertical, and 
hence the expression for them is 23. 
i¢: 10: 1a. Inthe annexed figure -a 
the two bisecting lines, a —a and 
b —D, represent the lateral axes; 
the line s ¢ stands for a section of 
a lateral plane of the first ofthese -b o Oem OE ee 
prisms, it being parallel to one ee 
lateral axis and meeting the othez 
at its extremity, and ad for that LE t 
a 
of the other, it meeting the two 
at their extremities. 
In the eight-sided prisms (figs. 14, 15), each of the lateral planes is 
parallel to the vertical axis, meets one of the lateral axes at its extrem- 
ity, and would meet the other axis if it were prolonged to two or three 
or more times its length. The line ag, in fig. 23, has the position of one 
of the eight planes; it meets the axis 0 at 0, or twice its length from 
the centre ; and hence the expression for it would be 7c: 20: 1a, or, 
since 6 =a, ic: 2: 1, which is a general expression for each of the eight 
planes. Again, ap has the position of one of the eight- planes of an- 
other such prism ; and since Op is three times the length of Ob, the ex- 
pression for the plane would be 7¢:3:1. So there may be other eight- 
sided prisms; and, putting » for any possible ratio, the expression 
ie: nm: 1is a general one for all eight-sided prisms in the dimetric sys- 
tem. 
A plane of the octahedron of fig. 16 meets one lateral axis at its 
extremity, and is parallel to the other, and it meets the vertical axis ¢ 
at its extremity ; its expression is consequently (dropping the letters a 
and 6, because these axes are equal) l¢:¢:1. Other octahedrons in 
_ the same vertical series may have the vertical axis longer or shorter 
than axis c; that is, there may be the planes 2c¢:7:1, 3¢:7:1, 
4¢:%¢:1, and so on; or 4¢:%:1,4¢:7:1, and so on; or, using m for 
any coefficient of c, the expression becomes general, mc:2:1. When 
m = 0 the vertical axis is zero, and the plane is the basal plane O of 
the prism ; and when m = infinity, the plane is 7¢: 7: 1, or the vertical 
plane of the prism in the same series, 7-2, fig. 10. 
The planes of the octahedron of fig. 17 meet two lateral axes at their 
extremities, and the vertical at its extremity, and the expression for the 
plane is hence 1¢: 1:1. Other octahedrons in this series will bave the 
general expression mc:1:1, in which m may have any value, not a 
decimal, greater or less than unity, as in the preceding case. When in 
this series m = infinity, the plane is that of the prism zc: 1: 1, or that 
of fig. 12. 
In the case of the double eight-sided pyramid (figs. 20, 21, 22), 
the planes meet the two lateral axes at unequal distances from the 
centre; and also meet the vertical axis. The expression may be 
2 
