89 
OI in E, and OS in F, and FG be drawn parallel to OJ meeting 
the plane JOK in G, we shall have 
eu. OH = (ew + fut gw). FG. 
But OSs Or — SE FG: 
Therefore 
(eu+ fu-+ ow). OF : (pu+qu+rw).OS=e.0H : p. OL. 
Hence if the ray kJ meet planes parallel to the zone-planes 
efg,pqr in the points D, Q, we shall have 
(eh +fk+gl).OD : (pht+qk+rl).0Q=e¢.0H : p. OL 
Therefore 
eut+futgw OF pu+qut+rw OS 
eht+fkt+el OD ph+qk+rl OQ- 
From this equation the anharmonic ratio of four rays in one 
zone-plane, of four zone-planes intersecting one another in one 
ray, and the indices of rays and zone-planes when the axes are 
changed, may be found as in (12), (18), (14), (15). 
PROPERTIES OF A SYSTEM OF POINTS ON THE SURFACE OF 
A SPHERE. 
Poles. 
20. Let the surface of a sphere having its centre in the 
origin of the system of rays meet the rays 100, 01 0, OOL, LLU 
in A, B, C, G, and the ray hkl in P, fig. 1. Let the great 
circle AP meet the great circle BC in L. From any point & in 
OP draw RQ parallel to OA meeting OL in Q. Draw QN 
parallel to OB meeting OC in N. Then (7), since QR, NQ, ON 
are the values of a, y, 2 at LR, 
Oi NG LON 
hee RS) Fin 
But 
ON : NQ=sin BL: sin CL =sin ABsin BAP : sinCA sin CAP. 
