84 
normal to OK, meet the zone-planes efg, pqr in Op, Or; KOQ, 
KOS in Og, Os; and planes through DF, QS parallel to OK, 
in df qs. Then df, qs will be parallel to Op, Or; 
Od: Oq=OD : OQ; and Of: Os=OF: OS. 
Therefore sin pOg: sin pOs=sind : sin f= Of : Od, 
and sin rOg : sin rOs = sing : sins = Os : Og. 
u sin pOgsinrOs eh+fk+gl put+quv+rw 
ence ———-* — = : , 
sin pOUssinrOg eu+ty+gw pratqk+rl 
where KOP, KOQ, KOR, KOS are four zone-planes inter- 
secting one another in one ray; efg, pqr the symbols of 
KOP, KOR; hkl, wvw the symbols of rays contained in the 
zone-planes KOQ, KOS; pOgq, pOs the angles which KOP 
makes with AOQ, KOS; and rOg, rOs the angles which KOR 
makes with AOQ, KOS. 
Since the order of the zone-planes KOP, KOQ, KOR, 
KOS is the same as that of the rays OP, OQ, OR, OS, it 
follows from (11) that the expression which forms the right- 
hand side of the preceding equation is positive except when one 
only of the zone-planes KOP, KOR lies between the other 
two. 
Indices of a ray when the axes are changed. 
14, Let planes parallel to the zone-planes efg, hkl, pqr 
meet the ray mno in D, L, Q, and the ray uvw in FN, 8. 
Then (11) 
eut+futgwOF  hu+kv+lwON _ pu+qu+rw OS 
em+in+go OD hn+kn+lo OL” pm+qn+4 ro OQ” 
Let the zone-planes hk], pqr intersect in the ray OA’; 
the zone-planes pqr, efg in the ray OB’; and the zone-planes 
efg, hkl in the ray OC’. And let m’n'o’, uv’ w’ be the 
symbols of the rays OQ, OS when referred to the rays OA’, 
OB’, OC’ as axes! The symbols of the zone-planes efg, hk], 
