80 Mr DAVIES on the Nature of the Hour-Lines 



THE EQUATIONS OF THE HECTEMORIAL CURVES 

 UPON THE SURFACE OF THE SPHERE 



I. 



REFERRING the curve to great circle co-ordinates, originating at the 

 elevated intersection of the meridian of the place with the equator, and esti- 

 mated upon these circles ; and moreover putting 



I = inclination of the horizon and equator, or co-latitude of the place for which 



the dial is made ; 



D = declination of any one of the corresponding semi-diurnal arcs ; and 

 L = longitude of a point in one of these curves whose declination is D ; 



then, by right angled spherical triangles, 



sin ~ tan D cot I = ascensional difference ; or complement of the semi-diurnal 

 arc ; and therefore 



90 sin" 1 tanD cot I 



JL = or 



n 



tan D = tan I cos L 



( A D,L). 



Such is the general equation of the hectemoria upon the spherical sur- 

 face, a few properties of which we proceed to examine, in order to furnish 

 the means of ascertaining whether it be a circle or not. 



II. 



The curve is continuous, or the dial-lines in question form, generally, 

 only a part of the curve whose equation we have just deduced. This will 

 be apparent from the form in which it involves cos L, a quantity which 

 will vary incessantly as L is made to vary. We shall trace it through one 

 of its systems of changes. 



