514 Prof. Miller^s Crystallographic Notices. 



HDE, GED, HED. From the symbols of D, E, G, H find the 

 symbols of the zone-circle DE, of one of the zone-circles DG, 

 DHj and of one of the zone-circles EG, EH. Then, the sym- 

 bols of the zone-circles through D and two of the three poles 

 E, G, H, the symbol of the third pole, and that of P, and the 

 angles GDE, HDE being known, the angle PDE is given by (/3). 

 In the same manner the angle PED may be found. Hence, 

 knowing the arc DE, and the angles PDE, PED, the arcs PD, 

 PE may be found, and the position of P determined with respect 

 to D and E. 



When any other pole T occurs in the zone-circle DP, the sym- 

 bol of the pole in which DP meets either EG or HG, and its 

 distance from D, may be found, and then DT is given by (a). 



Nearly all the expressions for calculating the dihedral angles 

 of crystals given in the new edition of Phillips's ' Mineralogy,' 

 are merely particular cases of («) and (yS). These very simple 

 and useful formulae cannot be readily expressed in any notation 

 which differs essentially from the notation used in the work 

 above mentioned ; and cannot be expressed at all in any notation 

 which merely designates a simple form, without being able to 

 distinguish from each other the different faces of which it is 

 composed. 



To find the direction of the axis of a zone. 



The determination of the direction of the axis of a zone in 

 terms of the indices of the zone is important, inasmuch as it 

 materially aids us in forming a distinct conception of the nature 

 of crystalline forms, and is also useful in constructing models of 

 crystals and in drawing their figui-es. In the Philosophical 

 Magazine for May 1857, I determined the direction of the axis 

 of a zone by means of elementary geometry. The following in- 

 vestigation was undertaken in order to render more complete 

 the treatment of Crystallography by spherical trigonometry. 



Let the axes of the crystal OX, 

 OY, OZ meet the surface of the 

 sphere of projection in X, Y, Z. 

 Let a, b, c be its parameters ; A, 

 B,C the poles of 10 0, 010,001; 

 K the pole of the zone- circle uvw. 

 P, Q, R the points in which the 

 zone-circle uvw intersects BC, CA, 

 AB. XKL, YKM, ZKN great 

 circles meeting YZ, ZX, XY in 

 L, M, N. 



Observing that the great circles 

 RX, RY, QX, QZ, PY, PZ make with PC, QC, RB and PQR, six 



