66 ON THE ANTIQUE HOUR-LINES. 
Liat 0 sp : : 
y = sin. G5 tan. pol. dist. in this equation, when the describing 
diameter arrives at H, sin. g°= 0, that is, the curve cuts the 
axis of the absciss2 at H, and when the describing diameter 
‘ ee : 
passes onward, the sign of e538 changed, expressing that there: 
are two similar and equal branches, one on each side of the 
axis of the abscissz. 
n ee 
Cos. 6° has only one value, because g ° 18 never so great 
as 90°, and its cosine, therefore, does not Recore =9@ 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 mci two astro~ 
ifaraieal 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 ; the rectilinear asymptotes with- 
in. 
