1 12 Mr DAVIES on the Nature of the Hour-Lines 



The same result has, however, been already obtained (XV.) by means 

 of the central projection of the general equation upon the polar tangent 

 plane. For it is obvious that tan D' of that equation is the r of this ; and 

 that L of that is ? of the present. The two equations 



tan 1)' cot I sec n L 

 r = cot I sec n >' 



evidently differ then in nothing but the notation. 



But, independently of the general equation on the sphere, or its pro- 

 jection, the equation of these curves upon the polar tan- 

 gent plane might have been readily found. For, let A be 

 the pole, D the intersection of the meridian with the ho- 

 rizon, and BGC the projection of any semidiurnal arc. 

 Let also E be one of the hectemorial points in that circle. 

 Also let a = radius of the sphere, and r AE. 



AD =r cot I 



a cot I /~i A /-i /-i A /-I i # cot I 



- = cos GAC, or GAC = cos" 1 - . 



^**^\ fi co\ T 



Hence GE = ^= cos" 1 - ; 



r = a cot I sec n V. 



; or, finally, by reduction, 



XXVI. 



The equation employed by Mr CADELL certainly shews that the locus 

 is not, generally, a straight line ; and, therefore, that on the sphere the 

 hectemoria are not great circles: but he does not attempt to prove by 

 means of it that the locus is not a conic section, and that therefore the curve 

 on the sphere cannot possibly be a less circle. I say " possibly," for we 

 should not be justified in the inference, that because the projection was a 

 line of the second order, the curve itself was, therefore, a less circle, 

 an oversight that has been inadvertently made by more than one respectable 

 geometer. The curve of penetration of a sphere with a concentric cone of 



