82 
The intersection of any two zone-planes 1s a ray. 
9. The zone-planes hk], pqr have for their equations 
no 4k 2412 =0, and p-tqZ+r-=0. 
a \iueiipe Pia Gr BB laey 
The equations to the intersections of these planes will be 
where u=kr—lq, v=lp—hr, w=hq—kp. 
The quantities w, v, w are obviously integers, and therefore 
the intersection of any two zone-planes is a ray. 
Condition that a ray may he in a zone-plane. 
10. Let the zone-plane pqr contain the ray www. The 
equations to the zone-plane and ray are 
BALE 
uz vB wy” 
Therefore, since the plane contains the ray, 
x y Zz 
—+q5+r-—=0, and 
Bae ay 
putqut+rw=0. 
Portions of two rays cut off by parallels to two zone-planes. 
11. In fig. 2 let OQ, OS be the rays hkl, wvw, and let 
the zone-plane QOS intersect the zone-planes efg, pqr, mm 
OP, OR. Let the planes having for their equations 
a he By heehee eae ee 
and therefore parallel to the zone-planes efg, pqr, meet the 
ray hkl in D, Q, and the ray wvw in FS. Let planes 
passing through D, Q, F, S, parallel to the zone-planes 1 0 0, meet 
the ray 100 in d,qg,f,s. Then Od, Og, Of, Os will be the 
values of x at the points D, Q, F, S. Therefore, since the 
equations to the rays hk/, wv w are 
ee eae aan ie Oh 
ha kB ly’ aug uz vB wy’ 
