ON THE ANTIQUE HOUR-LINES. 



67 



in which the curves are contained, are the ninth and fifteenth 

 astronomical hour-line counted from midnight. 



From figure 1st it may be collected, that the asymptotes of 

 each hectemorial line are as foll6ws : 



The equicrural curvilinear branch, 

 composed of the hectemorial lines 



First A, and Eleventh IA, 

 Second B, and tenth I, 

 Third I\ and ninth 0, 

 Fourth A, and eighth H, 

 Fifth, E, and seventh Z, 



has for asymptotes the astro- 



which compre- 



nomical hour-lines, counted 



hend an angle 



from midnight, 



of 



VII. and XVII. 



150° 



VIII. and XVI. 



120° 



IX. and XV. 



90° 



X. and XIV. 



60 '> 



XL and XIII. 



30° 



— 1 



Figure 1st represents the portions of the day hectemorial 

 lines which touch the greatest invisible parallel for the latitude 

 66° 30' ; these are the winter portions. In order to delineate 

 the portions which touch the greatest wholly visible parallel, 

 or the summer portions, the projection is made on the plane 

 which touches the sphere at the depressed pole ; this projec- 

 tion is exhibited in figure 2d, which contains portions of 

 curves, each of which is similar and equal to the curve of the 

 same hour in figure 1st, but differently placed. It forms a su- 

 perior equinoxial dial for 66° 30', when placed parallel to the 

 equator, with the inscribed surface upwards, and the point xxiv. 

 elevated. 



However extended the plane of projection touchino- the 

 sphere at the pole be, still it will not contain the portions of 

 the hectemorial lines that are near the equator. To have these 

 the plane of projection is taken at right angles to the equator, 

 and touching the sphere at the depressed intersection of the 



I 2 equator 



