Mr DAVIES on the Nature of the Hour- Lines 



after certain intervals corresponding thereto, and fall upon the original se- 

 ries, ad infinitum. 



If n be a proper fraction, the quantity L must be greater than the semi- 

 diurnal arc, and the curve will pass into the nocturnal portion of the sphere, 

 or proceed beyond and rise up again on the diurnal portion. These posi- 

 tions may be fixed after the occurrence of several revolutions, depending, as 

 is too obvious to need further specification, upon the value of the fraction n. 



VI. 



The question then, at issue, is decided. The hectemoria are not 

 circles ; for the circle has not one property in common with those we 

 have, shown to characterise this class of curves. 



VII. 



The species of a curve depends upon the relation amongst its constants ; 

 the order upon the relation among its variables. It often happens, how- 

 ever, that specific relations amongst the constants, affects also the order 

 of a curve ; and whenever it does so, it is by depriving it of its highest 

 terms, or by destroying all those which do not contain a variable factor com- 

 mon to the whole. This circumstance takes place in the case before us ; for 

 though, generally, the curve upon the sphere is of a higher order than the 

 circle, and consequently is a curve of double curvature, yet, in particular 

 cases, it becomes a great circle of the sphere. It never, however, becomes 

 a less circle. The general equation (A DL ), and its modifications, deter- 

 mine these cases with great simplicity. 



1. Take L = cos" 1 tan D cot I 



n 



If, iii this respect, I = 90, then cot 1=0, and 



L = = , for every point in the curve. 



n n 



Hence, these hectemoria are the equinoctial hour-lines of the sphere, 



