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 lateral axis, and meet the other lateral axia 

 at its extremity. The expression for it is hence (c standing for the 

 vertical axis and a, b for the lateral) ic : ib : la, i, 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. 



a : lb : la. In the annexed figure 

 the ko bisecting lines, a —a and 

 b ~-b, represent the lateral axes ; 

 the line s t stands for a section of 

 a lateral plane of the first of these 

 prisms, it being parallel to one 

 lateral axis and meeting the other 

 at its extremity, and ab for that 

 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 ao, in fig. 23, has the position of one 

 of the eight planes ; it meets the axis b at o, or twice its length from 

 the centre ; and hence the expression for it would be ic : 2b : la, or, 

 since b = 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 ic : 3 : 1. So there maybe other eight- 

 sided prisms ; and, putting n for any possible ratio, the expression 

 ic : n : 1 is 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 G 

 at its extremity ; its expression is consequently (dropping the letters a 

 and b, because these axes are equal) lc : i : 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 2e : i : 1, 3c : i : 1, 

 4.C : i : 1, and so on ; or \c : i : 1, \c : i : 1, and so on ; or, using m for 

 any coefficient of c, the expression becomes general, mc .: i : 1. When 

 m = the vertical axis is zero, and the plane is the basal plane of 

 the prism ; and when m = infinity, the plane is ic : i : 1, or the vertical 

 plane of the prism in the same series, i-i, 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 \c : 1 : 1. Other octahedrons in this series will have 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 ic : 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 



