87 
“0U=%0V=~ OW, 
Brgy yy, 
where u=kr—lq, v=lp—hr, w=hq—kp. 
A plane through the rays hkl, pqr, and therefore parallel 
to the plane UVW, will be called a zone-plane, and will be 
denoted by the symbol u v w, or by any three integers respect- 
ively proportional to u, v, w; and the integers u, v, w, or any three 
proportional integers, will be called the indices of the zone-plane. 
The intersection of any two zone-planes 1s a ray. 
18. In OB, fig. 6, take OB=. Let planes through B 
parallel to the zone-planes h k |, p q r intersect one another in - 
the line BM which meets the plane COA in M; and let them 
meet OC in L, R, and OA in H, P. Then 
DH Ob Ol and” OR OB aOR. 
a B y a B Y 
Therefore 
1.0L=ky, h. OH=ka, r. OR=qy, p.OP = qa. 
Hence 
Ir. LR=(kr—1q)y, hp.HP = (hq —kp)z. 
But (Tract 187) 
HM.OP.LR=HP.OR. LM. 
Therefore, putting 
u=kr—lq, v=lp—hr, w=hq—kp, 
we have : 
wl. LM=uh. HM, wl. LH =—1k. HM, uh. LH=—vk. LM. 
Draw MD parallel to OC meeting OA in D. By similar 
triangles OD : LM=OH : LH, and DM: HM=OL : LH. 
Hence —v.OD=u2, and —v.DM=wy. Draw MF equal and 
parallel to OB on the opposite side of the plane LOH. Then 
—v.MF=v.O0OB=vr. The line OF being parallel to BM, is 
