Elementary Geometry to Crystallography. 849 



once from those of h, k, I, p, q, r without the trouble of substi- 

 tuting the latter in the expressions for u, v, w. 



8. The planes of a system that intersect in parallel lines are said 

 to constitute a zone. A line through the origin parallel to the 

 edge formed by any two planes of a zone is called the axis of the 

 zone. A zone and its axis are denoted by the symbol of the 

 edge formed by any two planes belonging to the zone. 



Let uvw be the symbol of a plane parallel to the edges hkl, 

 pqr. Then M = kr— Iq, v = lp — hr, t<;=hq— kp. Therefore, 

 multiplying the first equation by h, the second by k, and the 

 third by 1, and adding, we get 



hM + ky + lw=0, 

 which expresses the condition that the zone hkl may contain the 

 plane uvio. 



Any three whole numbers which, when substituted for u, v, w, 

 satisfy the above equation, are the indices of a plane in the zone 

 hkl ; and any three whole numbers which, when substituted for 

 h, k, 1, satisfy the same equation, are the indices of a zone con- 

 taining the plane uvw. Fig. 5. 



9. Let the plane uvw meet the axes 

 of the system of planes in U, V, W, 

 and the zone-axis efg in P. Draw 

 VP meeting WU in M, and WP 

 meeting UV in N. The indices of 

 the edge VM will be —ev, gw + ew, 

 — gv, and the indices of the edge 

 WN will be — ew, —iw, eu + iv. 

 The edges VM, WN are in the plane 

 UVW. Therefore by (5), 



CM . WU = (eM + gw)WM, and fw . VN=eM . NU. 

 But by (2), PV . NU . WM=MP . VN . UW. Therefore 

 fy.PV=(eM + gM;)MP. 



fi;.MV=(eM + fi;-fg«;)MP. 



In like manner, if the plane mno passing throiigh V, meet OP 

 in D, and OM in G, 



fn.GV=(em + f« + go)GD. 

 But by (2), OD . GV . MP=OP . GD . MV. Hence 

 v{c,m + fn + go)OD = n(eu + io + gw)OP. 



So also, if the zone-axis hkl meet the plane uvw in Q, and the 

 plane mno in E, 



i;(hOT-|-k/i-f-l«)OE=n(hM + kv-FlM;)OQ; 



