86 
~ GEOMETRICAL INVESTIGATION OF THE PROPERTIES OF A 
SYSTEM OF RAYS. 
Rays. 
16. In fig. 4 let O be the origin of a system of rays; OA, 
OB, OC the rays 100, 010,001; OR the ray hkl; HKL a 
parallelopiped having its edges in OA, OB, OC, and having 
Of for a diagonal. Then (7), since OH, OK, OL are the values 
of 2, y, z for the point R, we shall have 
OH _OK _OL 
ee TES 
Zone-planes. 
17. In OA, fig. 5, take OU =—a, and therefore measured 
from O in the direction opposite to A. Through U draw UW, 
US parallel to the rays hkl, p qr respectively, meeting the 
plane BOC in M,8S. Let MS meet OB in V, and OC in W. 
Draw MD, SG parallel to OC meeting OV in D, G. The 
lines UM, US are parallel to the rays hk, pqr, therefore, 
observing that smce OU =—a, UO=a, 
T0_OD_DM ,, YO_0G_as 
(agi TGR pe Ge ry 
Hence 
k l q 0G 
OD=-8, DM=-y, OG=24B, GS=-y. 
Z 7,7 Be wi 
The lines DM, GS are parallel to OW, therefore 
OW:0V=DM: DV=DM-—GS: OG—OD; 
consequently 
(hq — kp) 8.OW = (Ip — hr)y.OV, h (Ip — hr). DV = (hq — kp), 
(Ip—hr).OV =(lq—kr)B8, (hq — kp). OW = (lq — kr)y. 
Hence, if a plane parallel to the rays hkl, pqr meet 
OA, OB, OC in UVW, 
