88 
the intersection of the zone-planes h k ], p q r, and is evidently 
the diagonal of a parallelopiped the edges of which are respect- 
ively coincident with the axes of the system of rays, and equal 
to OD, MF, DM, and therefore proportional to 
—v. OD, —v. MF, —v.DM, or to uz, vf, wy. 
Since w, v, w are integers, the line OF, in which the zone- 
planes hk], p qr intersect, is a ray of the system, having uv w 
for its symbol, where 
w=kr—lg, v=lp—hr, w=hq —kp. 
Portions of two rays cut off by parallels to two zone-planes. 
19. Leta plane parallel to the zone-plane p q r meet the 
axes in I, J, K, fig.7, and the ray wvw in S. Draw KS meeting 
IJ in N, IS meeting JK in L, and ST parallel to OJ, meeting 
the plane JOK in T. The symbols of the rays OK, OJ, OS 
are 001, 100, wow respectively. Therefore the symbol of 
the zone-plane KOS will be v w 0, and that of the zone-plane 
LOS will be Ow. The plane IJK is parallel to the zone-plane 
pqr. Hence the line AWN will be parallel to the ray 
Se UU SS Tele 
and the line JZ will be parallel to the ray 
qu+rw, —pv, — pw. 
The lines KN, IL are in the plane IJK, therefore (18) 
pu.IN=quv.JN, and qu. JK=(qu+rw). KL. 
But (Tract 187) JESS TEGD) SUN SIG AEG, JONG 
Hence pu. IS =(qv + rw). SZ, 
and therefore pu.IL = (pu + qu+rw) SL. 
But 8 lie LOUISE = eile. 
Therefore pu. OL= (put qut+rw). ST. 
In like manner, if a plane parallel to the zone-plane efg meet 
