92 
These equations are satisfied by making 
wu=kr—lq, v=Ip—hr, w=hq—kp. 
It is evident that wu, v, w are integers, therefore P is the 
pole wv w, and P’ is the wow. 
Relation between the arcs AK, BK, CK, K being a pole of the 
zone-circle EF. 
24. Let uvw be the symbol of the zone-circle EF, and 
let K be the pole of the zone-circle EF nearest to C. Then 
cos AK =—sin AH'sin EF, cos BK=—sin BDsin D, 
cos CK =sin CD sin D=sin CE sin E. 
The symbol of D is 0 wv, and the symbol of H is w0u. There- 
fore (20) 
csAK  smAk _ sin AB sin ABE _ uy 
cosOK  snCEk sin BO sin CBE wa’ 
cosBK sn BD sn AB sin BAD _ vy 
csCK snCD  sinOA snCAD wf’ 
Whence 
and 
= = cos AK=& cos BK = + ~ cos CK. 
Anharmonic ratio of four zone-circles passing through one 
pole. 
25. In fig. 9 let A, B,C be the poles 100,010, 001 
respectively; AP the zone-circle e f g intersecting the zone- 
circles CA in M; KR the zone-circle p qr intersecting the zone- 
circle BC in N; Q the pole hkl; S the pole wvw. Let the 
zone-circles K@, KS intersect the zone-circle MN in 7, V; also 
let €7 0, 6y w be the symbols of 7, V respectively. Then 
(21), (23) the symbol of M will be g0e, the symbol of N will 
be 0 rq, the symbol of MN will be er gq gr, the symbol of K 
will be fr — gq gp — er eq — fp, and the symbol of KQ val be 
keq—kfp —lgp + ler, 
lfr —lgq—heqt+hfp, 
hep—her—kfr +kegq. 
