66 



ON THE ANTIQUE HOUR-LINES. 



y — sin. g 5 tan. poL dist. in this equation, when the describing 



n 



diameter arrives at H, sin. ^ s . = 0, that is, the curve cuts the 

 axis of the absciss e at H, and when the describing diameter 



71 



passes onward, the sign of - s is changed, expressing that there 



are two similar and equal branches, one on each side of the 

 axis of the abscissas. 



77 /m 



Cos. - s has only one value, because - 



6 



n s is never so great 

 6 ° 



as 90°, and its cosine, therefore, does not become rr 0, which 

 it must do before its sign changes. 



If a central projection of any one forenoon hectemorial hour- 

 line, and of the afternoon hectemorial line equi-distant from 

 the meridian, be drawn for 

 different heights of the pole, 

 it will be seen that these two 

 hectemorial lines form for 

 each height of the pole an 

 equicrural curvilinear branch, 

 including within it the cor- 

 responding hectemorial lines 

 for a higher latitude, and all of them included within two astro- 

 nomical hour-lines as asymptotes. With these asymptotes the 

 pair of hectemorial lines coincides, when the poles are in the 

 horizon. The figure in the margin represents the equicrural 

 curves formed by the third hectemorial hour-line and by the 

 ninth, which is the afternoon branch equi-distant from the me- 

 ridian. 



The curved branches are drawn for every ten degrees of la- 

 titude from 30 to 70 inclusive j the rectilinear asymptotes with- 



in 



