104 Mr DAVIES on the Nature of the Hour-Lines 



The parallel of declination bound- 

 ing the hectemoria on the hemisphere 

 of contact is QR, the equator is F G ; 

 and the parallel of declination bound- 

 ing the hectemoria on the other hemi- 

 sphere, is projected upon the opposite 

 cone, and is developed into Q'R'. The 

 negative branches of the developed 

 curve, were, prior to development, si 

 tuated upon the opposite cone also. 



ON THE ANALYTICAL EQUATIONS OF THE HECTEMORIA. 



We have seen that, on the sphere, the equations of the hectemoria in- 

 volve a multiple cosine, and we may infer from the first principles of ana- 

 lytical geometry, that the same function must appear in all possible projec- 

 tions of their loci ; with this addition, however, that the expression might 

 be expected to assume, in general, a more complicated form. Still there is 

 one process, and perhaps one only, by which an expression can be obtained 

 from which the points of the projected curves can be computed with tole- 

 rable facility, more especially, if aided by one or two subsidiary tables. To 

 proceed systematically, I shall commence by finding the equation of the 

 hectemorial cone, concentric with the sphere, and referred to rectangular 

 co-ordinates ; and then proceed to determine its intersection with any plane 

 whatever. 



XXI. 



The Equations of the Hectemorial Cone. The spherical functions of 

 latitude and longitude may be readily transformed into rectangular ones by 



