93 
Hence (23) 
€=g (gq —fr) (ph + gk + 1), 
n=r (er — gp) (eh + fk + gi). 
In like manner 
$= g (gq — fr)(pu + qu + rw), 
y=r (er —gp) (ew+ fo + gw). 
sin PKQ sinRKS_ sin MT sin NV 
But on RKQ smPKS sin NT sin MV 
_ sin ACT sin BCV _ > 
~ sin BOT sin ACV. € va 
Therefore 
sin PKQ sin RES eh +th+gl pu+qu+rw 
sin PKS sin REQ eut+fu+gw ph+qk+rl — 
Anharmonic ratio of four poles in one zone-circle. 
26. Let the zone-circle QS meet the zone-circle KP in P, 
and the zone-circle KRin R. Then, since the anharmonic ratio 
of the points P, Q, R, S is the same as that of the ares AP, KG, 
KR, KS (Tract 16), 
sin PQ sin RS  eh+fk+eol put+qu+rw 
sin PS sn RY eu+fv+gw ph+qk+rl © 
eon of the expression forming the right-hand side of the final 
equations in (25) and (26). 
27. In (25) the left-hand side of the final equation may be 
replaced by its equivalent 
cot PKS — cot PKR 
cot PKQY— cot PKR’ 
and in (26) it may be replaced by 
cot PS — cot PR 
cot PQ — cot PR © 
From the form of these expressions it 1s manifest that they 
are positive, and therefore also the expression forming the right- 
hand side of the equations in both cases, except when one only 
of the zone-circles KP, KR lies between Q and 8. 
[Reprinted, 1880. ] 8 
