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 abscissae x be taken on the meridian HM, 

 and the ordinates y at right angles to the meridian, the cen- 

 tral projection gives 



n . ,. 



y zz sin. ^ s tan- polar dist. star, 



x— j cos. ^ s — cos. s >■ tan.pol. dist. star. 



.11 

 PH is cos. s ; the cosine of the semidiurnal arc. PM is cos. 7. s ; 



o 



the cosine of the fractional part of the semidiurnal arc. MS 



is sin. xs; 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, n is one of the numbers 1, 2, 3, 4, 5; it is 5 for 

 the curve that contains the first and eleventh hectemorial line ; 

 4 for the second and tenth, and so forth. 



. n 



y $m '6 



is not the equation of a straight line, except 



cos. „ s — cos. s 

 o 



in the case where cos. 5 = 0, that is, when the semidiurnal arcs 

 are all 90°, the poles being in the horizon. 



Vol. VIII. P. I. I y - 



