ON THE ANTIQUE HOUR-LINES. 69 
the first. The extremities of the diameter, after having in this 
way passed several times to and fro between the greatest whol- 
ly unseen and the greatest wholly seen parallel, aud after ha- 
ving completed one or more circumferences, attain precisely 
the two opposite points from which they set out, having form- 
ed two opposite re-entering curves on the surface of the sphere, 
and afterwards, in every subsequent revolution, the extremities 
of the diameter only retrace the lines they had described in 
their first revolution. The nature of the motion is such, that 
the describing points cannot go beyond the two parallels which 
touch the horizon, because there are no semidiurnal arcs be- 
yond these parallels, and the constitution of the hectemorial 
lines, consists in cutting the semidiurnal arcs. 
Whilst the extremities of the diameter describe two re-en- 
tering curves on the surface of the sphere, the diameter itself 
describes two opposite re-entering curved surfaces, whose com- 
mon vertex is the centre of the sphere. These two opposite sur- 
faces coincide with a straight line directed to the vertex, but 
do not coincide with a straight line in any other direction. In 
this respect they resemble a conical surface, and they may be 
considered as a kind of opposite cones with an undulated sur- 
face, included between the outer surfaces of the two opposite 
circular based cones that touch the horizon; the apex of each 
undulation being applied alternately to the one and to the 
other of these right circular based cones, so that as the com- 
mon vertex of all these cones is the centre of the sphere, each 
_of the two opposite undulated cones has two similar and equal 
_ bases or right sections, the one above the centre of the sphere, 
_the other below it. Each of these bases is an uninclosed curve, 
- composed. of many equiecrural branches, each equicrural branch 
haying for asymptotes two straight lines that intersect in the 
»axis ; these two right sections are indefinitely extended ; and 
the 
