326 Prof. Miller’s Crystallographic Notices. 
The angle which the distance between the centres of any two of 
the circles subtends at K, is equal to the angle between the 
corresponding originals. Therefore, since the anharmonic ratio 
of P, Q, R, S is the same as that of K P, KQ, KR, KS, 
PQ RS _ eht+fk+gl put+qut+rw T 
RQ PS” pht+gk+rl eutfv+gw- x 
Let the zone-circle Q S meet the 
zone-circles K P, KR in the poles 
P, R. Let T be the centre of the 
projection of the zone-circle QS; ¥ Q R 8 
TP, TQ, TR, TS the loci of the centres of projections of great 
circles the originals of which pass through P, Q, R, S respectively. 
Therefore, since the anharmonic ratio of the lines T P, TQ, TR, 
T'S is the same as that of the points P, Q, R, 8, 
sn PTQ sm RTS _ eh+fk+gl put+qu+rw 
sin RTQ sin PTS” ph+gk+rl eut+fo+gw’ 
The symbol of any zone-circle may be used to denote the 
centre of its projection, and the symbol of any pole may be used 
to denote the straight line which is the locus of the centre of the 
projection of a great circle passing through it. 
Let D, E, F, G be the centres of the projections of four zone- 
circles, no three of which are in one straight line; H the inter- 
section of DE, FG; M the intersection of the circles having 
their centres in D, E; N the intersection of the cireles having 
their centres in F, G. The straight line D E is the locus of the 
centres of the projections of great circles passing through the 
original of M; FG is the locus of the centres of the projections 
of great circles passing through the original of N. Therefore 
H is the centre of the projection of the great circle which is the 
original of MN. Hence, if the centre of the projection of a 
zone-circle be denoted by the symbol of the original, and the 
line joining any two centres be denoted by the symbol of the 
pole in which the originals intersect, the rule for finding the 
symbol of a zone-circle from the symbols of two poles in it, or 
for finding the symbol of the pole in which two zone-circles 
intersect, from the symbols of the zone-circles, may be applied 
to find the symbol of a line from the symbols of two centres 
through which it passes, or to find the symbol of the intersection 
of two lines each of which joins two centres, from the symbols 
of the lines. 
The expression 
eh+fk+gl put+qvu+rw 
ph+qk+rl eut+fo+gqw’ 
where efg, pqgr are the symbols of two zone-circles K P, KR, 
