85 
p qr when referred to the new axes will be 1 0 0, 010,001 
respectively. Therefore (11) 
OF _v ON _w 
inl OD Wi OG 6 
Hence, comparing corresponding terms, two equations are 
obtained which are satisfied by making 
== 
On 
m =em+in+go, w=eu+fv+gu, 
n=hm+kn+lo, v =hu+kv+lwu, 
o=pm+qn+ro, w=pu+qu+rw. 
The coefficients of wu, v, w are integers, therefore w’, v’, w’, the 
indices of the ray OS when referred to the rays OA’, OB’, OC’ 
as axes, are also integers. Hence, the rays of the system are 
subject to the same law when referred to any three rays as axes, 
as when referred to the original axes. 
Indices of a zone-plane when the axes are changed. 
15. Let the rays efg,hkl,p qr meet a plane parallel to 
the zone-plane mn o in D, H#, F, and a plane parallel to the 
zone-plane uv win fi, S, 7. Then (11) 
uet+vf+woOR uh+vk+wl OS up+v¢+wr OT 
me+nf+ogOD mh+nk+ol0OH mh+nq+or OF’ 
Let m’n’o’, uv w be the symbols of the zone-planes 
parallel to DHF, RST, when referred to the rays e fg, hk l, 
pqr as axes. The new symbols of the rays OR, OS, OT will 
be 100,010,001. Therefore (11) 
a Oe ww oOs sw Ole 
m OD nn OE o OF 
Hence, comparing corresponding terms, two equations are 
obtained which are satisfied by making 
m=en+/n+go, uw =eu+fv+gw, 
n =fAm+ihkn+lo, vw=hu+ky+lw, 
o=pm+qn+ro, w=put+qvtrw. 
