ON THE ANTIQUE HOUR-LINES. 65 
hectemorial lines great circles, and in this case they coincide 
with the astronomical hour-lines, and pass through the poles 
of the equator. When the poles are thus in the horizon, all 
the three kinds of hour-lines coincide. When the pole is in 
the zenith, the second kind or horizontal, and the third or 
hectemorial hour-lines cease to be, because then the horizon 
does not cut any of the parallels into diurnal and nocturnal 
arcs. 
To express algebraically some of the above-mentioned pro- 
perties. Let the abscisse 2 be taken on the meridian HM, 
and the ordinates y at right angles to the meridian, the cen- 
tral projection g gives 
n : 
ay SIN. 63 tan. polar dist. star, 
se 
si Or OFT... 8 n ee . 
“= | COs. & S—COS. 8 i tan. pol. dist. star. 
es i hee P oy) 
PH is cos. ; the cosine of the semidiurnal arc. PM is cos. rae 
iv } + ‘ 
the cosine of the fractional part of the semidiurnal arc. MS 
Bee: : ; 
is sin. & 8; the sine of the fractional part of the semidiurnal 
arc, PS is tan. pol. dist. star ; the tangent of the polar distance ~ 
of the star. m is one of the numbers 1, 2, 3, 4, 5; it is 5 for 
the curve that contains the first and eloton tie hectemorial line ; 
4 for the second and tenth, and so forth. 
it n 
sin. = 
vie hes is not tl tion of a straight li 
- s not the equation of a straight line, except 
“COS. 6 7. 5—COS. Sik 
in the case where cos. s = 0, that is, whet the semidiurnal arcs 
are all 90°, the poles being in the horizon. 
Vou. VIL P. I. I go 
