83 
we shall have 
(eh+ fk + gl)Od=mhi, (ph+qk+rl) Og = nha, 
(eu+fv+gw)Of=mua, (putqu+rw) Os =nua. 
But Od: Og=OD: OQ, and Of : Os=OF: OS. There- 
fore 
ew+tv+gw OF _put+tqu+rw OS 
eh +fk+el OD ph+qk+rl 0Q° 
When only one of the rays OP, OR lies between OQ and 
OS, three of the points d, qg, f,s will be on one side of O, and 
the fourth on the other side. Therefore Of. Os: Og. Od will 
be negative. When OP, OR are both without the angle OOS, 
or both within it, the points d, q, f, s will either be all on one 
side of O, or two on one side and two on the other side, and 
Of . Os : Og . Od will be positive. Hence the expression 
eh+fk+eol put+tqu+rw 
eut+fut+tew ph+qk+rl 
will be positive except when one only of the rays OP, OR lies 
between OQ and OS. 
Anharmonic ratio of four rays in one zone-plane. 
12. Since DF, QS are parallel to OP, OR respectively, 
sin POQ : sin POS=sin D: sin F= OF: OD, 
and sin ROQ : sin ROS=sin Q : sinS= O08 : OQ. 
sin POQ sin ROS eh +fk+ ol put+qu+rw 
sin POS sin ROQ eu+fu+ew ph+qk-+ ri ’ 
Hence 
where OP, OQ, OR, OS are four rays in one zone-plane; ef g, 
pqr the symbols of zone-planes containing the rays OP, OR; 
and hkl, wv w the symbols of the rays OQ, OS. 
Anharmome ratio of four zone-planes intersecting one another 
im one ray. 
13. Retaining the notation of (12), let the zone-planes e f g, 
p qr intersect in the ray OK; and let a plane through 0, in fig. 8, 
