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, a.nd 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 thev 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 uri inclosed curve 

 composed of many equierural branches, each equicrural branch 

 having for asymptotes two straight lines that intersect in the 

 axis j these two right sections are indefinitely extended j and 



the 



