upon the Antique Dials. 87 



and I becomes the pole P. This agrees with fact ; for the semi-diurnal arcs 

 being all semicircles, the meridians, of course, n sect them all. 



2. Take the equation in its original form, viz. 



tan D = tan I cos L ; 



and let 1 = 0. Then tan D = for all values of L ; which indicates that 

 the hectemoria in question have all disappeared. There are, in truth, now 

 no semi-diurnal nor semi-nocturnal arcs, and therefore no hectemoria, the 

 equator being in the horizon. 



3. In the same equation, put n = 1 ; then 



i 

 tan D = tan I cos L. 



This is the equation of a great circle whose inclination to L is I ; in short 

 equation of the horizon itself. 



4. As we increase n, our curves approach the meridian more and more 

 nearly ; and when n is become infinitely great, we get 



T cos~ tan D cot I 



inf. 



Hence, so long as the numerator is finite, the value of L. is 



L = 0, 



which is the equation of the meridian. 



Now, for any finite number of revolutions, the numerator is finite ; 

 and when that number is infinite, the expression becomes indeterminate. 

 Hence the meridian is the only assignable locus when n is infinite. 



These cases of circular hectemoria have been already noticed, both by 

 Mr CADELL and M. DELAMBKE ; and they are probably the only ones 

 in which the curve becomes so simplified. 



M 



