2S4- CRYSTALLOGRAPHY. 



tained by adding 90°. If » = 1, then the ratio is 1 : 1, as in ACB, 



and each angle equals 45°, giving 135° for the inclination on eithei 

 adjoining cubic face. 



Again, if the angles of inclination have been obtained by measure- 

 ment, the value of ?i in any case may be found by reversing the above 

 calculation ; subtracting 90 ~~ from the angle, then the tangent of this 

 angle, or the cotangent of its supplement, will equal n, the tangents 

 varying directly with the value of ?i. 



In the case of planes of the m : 1 : 1 series (including 1:1:1. 2:1: 

 1, etc.), the tangents of the angle between a cubic face in the same 

 zone and these planes, less 90 ; , varies with the value of vi. In the 

 case of the plane 1 (or 1 : 1 : 1). the angle between it and the cubic face 

 is 125° 16'. Subtracting 90% we have 35° 16'. Draw a right-angled 

 triangle, OBO. with 35° 16' as its vertex angle. BC haa 

 the value of lc, or the semi-axis of the cube. Make 

 DC=2BC. Then, while the angle OBO has the value 

 of the inclination on the cubic face less 90" for the plane 

 1:1:1, ODC has the same for the plane 2:1:1. Now, 

 making OG the radius, and taking it as unity, BC is the 

 tangent of BO C, or cot OBC. SoDC = 2BC is the tan- 

 gent of DOC, or cot ODC. By lengthening the side CD 

 (= 2BC or 2c it may be made equal to hBC = 3c, its 

 value in the case of the plane 3:1:1; or to 4Z?C = 4c, 

 its value in the case of the plane 4:1:1; or mBC = mc 

 for any plane in the series m : 1 : 1 ; and since in all 

 there will be the same relation between the vertical and 

 the tangent of the angle at the base (or the cotangent of the angle at 

 the vertex), it follows that the tangent varies with the value of m. 

 Hence, knowing the value of the angle in the case of the form 1 

 (1:1:1), the others are easily calculated from it. 



BC being a unit, the actual value of OG is \ j/§, or \\, it being half the 

 diagonal of a square, the sides of which are 1, and from this value the 

 angle 35 : 16' might be obtained for the angle OBC. 



The above law (that for a plane of the m : 1 : 1 series, the tangent of 

 its inclination on a cubic face lying in the same zone, less 90 J , varies 

 with the value of m. and that it may be calculated for any plane 

 m : 1 : 1 from this inclination in the form 1:1:1), holds also for 

 planes in the series m : 2 : 1, or m : 3 : 1, or any m : n : 1. That is, 

 given the inclination of on 1 : n : 1, its tangent doubled will be tfiat 

 of 2 : n : 1, or trebled, that of 3 : n : 1, and so on; or halved, it will be 

 that of the plane £ : n : 1, which expression is essentially the same us 

 1 : 2/i : 2. 



These examples show some of the simpler methods of applying ma- 

 thematics in calculations under the isometric system. The values oi 

 the axes are not required in them, because a = b = c — 1 . 



3. Hemihedral Crystals. — The forms of crystals described 

 above are called holohedral forms, from the Greek for all and 

 face, the number of planes being all that full symmetry re- 

 quires. The cube has eight similar solid angles — similar, that 

 is, in the enclosing planes and plane angles. Consequently the 

 la^ of full symmetry requires that all should have the same 



