81 
h, k,l any three integers positive or negative or zero, one of 
them at least remaining finite; let OA, OB, OC be taken as 
co-ordinate axes; and let a system of lines be constructed by 
giving different values to h, &, J in the equations 
FS i 
ha kB ly’ 
That portion of any line defined by the preceding equations, 
which lies on one side of the origin, will be called a ray. The 
ray containing the point x=ha, y=k®, z=ly will be denoted 
by the symbol Akl. The ray containing the point «=— ha, 
y =—k8, z=—ly will be denoted by the symbol hkl. The 
integers hk 1 will be called the indices of the ray; and the lines 
a, 2, y will be called the parameters of the system of rays thus 
constructed. 
The lines Od, OB, OC are manifestly the rays 10 0, 
010,001. 
Zone-planes. 
8. The equations to the rays hkl,p qr are 
OO ae 
fi SO ea. and fon Of8 157 
Hence the equation to the plane containing the rays h k 1, 
pqr will be 
uety t4w =0, 
Ct aa 
where u=kr—lq, v=lp—hr, w=hq—kp. 
This plane will be called a zone-plane, and will be denoted 
by the symbol uv w. The quantities u, v, w will be called its 
indices. ‘They are evidently positive or negative integers one 
or two of which may be zero. 
The symbols of the zone-planes BOC, COA, AOB are 100, 
010, 001 respectively. 
