DIMETlvIC, OK TETRAGONAL SYSTEM. 35 



lations to the basal plane were known, then, after subtracting from 

 the values 90°, the cotangents of the angles obtained, or the tangents 

 of their complements, will equal m in each case ; that is, the tangents 

 (or cotangents) will vary directly with the value of m. The logarithm 

 of the tangent for the plane 1-i, added to the logarithm of 2, will 

 equal the logarithm of the tangent for the plane 2-i, and so on. 



The laio of the tangents for this vertical zone m-i holds for the planea 

 of all possible vertical zones in the dimetric system. Further, if ths 

 square prism were laid on its side so that one of the lateral planes be- 

 came the base, and if zones of planes are present on it that are vertical 

 with refe?ence to this assumed base, the law of the tangents still holds, 

 with onty this difference to be noted, that then one of the lateral axes 

 is the vertical. It holds also for the trimetric system, no matter which 

 of the diametral planes is taken for the base, since all the axial inter- 

 sections are rectangular. It holds for the monodinic system for the 

 zone of planes that lies between the axes c and b and that between the 

 axes a and b. since these axes meet at right angles, but not for that 

 between c and a, the angle of intersection here being oblique. It holds 

 for all vertical zones in the hexagonal system, since the basal plane in 

 this system is at right angles to the vertical axis. But it is of no use 

 in the triclinia system, in which all the axial intersections are oblique. 



The value of the vertical axis e may be calculated also from the incli- 

 nation of on 1, or of Jon 1. See fig. 2, and compare it with fig. 17. 

 If the angle / on 1 equals 140°, then, after subtracting 90°, we have 50° 

 for the basal angle in the triangle OGB, fig. 24 ; or for half the inter- 

 facial angle over a basal edge— edge Z— in fig. 17. The value of 6 

 may then be calculated by means of the formula 



c = tan \ Zvh 



by substituting 50° for \Z and working the equation. 

 For any octahedron in the series in, the formula is 



mc = tsiniZvi 



Z being the angle over the basal edge of that octahedron. If m = 2, 

 then c = i (tan \Zy/\). Further, m = (tan \Z v'%) -J- c. 



The interfacial angle over the terminal edge of any octahedron m 

 may be obtained, if the value of c is known, by the formulas 



mc = cot e cos e = cot %X 



X being the desired angle (fig. 17). The same for any octahedron m-i 

 may be calculated from the formulas 



mc = cot € cos e = cos %Yv2 



Y being the desired angle (fig. 16). 



For other methods of calculation reference may be made to the " Text 

 Book of Mineralogy," or to some other work treating of mathematical 

 crystallography. 



3. Hemihedral Forms.— Among the few hemihedral forms 

 nmler the dimetric system there is a tetrahedron, called a sphen* 



