98 Mr DAVIES on the Nature of the Hour-Lines 



Again, HQ = HG GQ = a (tan HOG tan DOG) 



( tan D tan X ) .. 



= a-{ cotX . [, or, finally, 



( 1 + tanDtanXJ' " 



a cosec 2 X _ a cosec 2 X cot I //-. 



"~ cot X + tan I cos wL ~ cotXcotl-f-coswL ^ D ' L 



XV. 



We might have estimated the values of v from the intersection of the 

 cone with the equator of the sphere, in directions tending to H. Thus, let 

 F be the intersection of this arete of the cone with the equator ; then, 



tf = FG + GQ = a\ tanx + 



' 



L + tan D tan X 



tan D sec 8 X _ sec X cosec X cos n L //-,, ^ 

 1 + tan X tan D ~ ' cot I cotX + cosL " D > L -' 



The chief difference between these two expressions, so far as utility is 

 concerned, is, that the value of v has a constant numerator, while that of v' 

 involves the variable factor coswL. 



XVI. 



If in the former of these equations (C D t ) we put x = 0, the result is, 



v = infinity ; 

 indicating that the origin or vertex of the cone is infinitely remote. 



But if we put \ = 90, then we obtain 



a cot I 



= 



the same equation that we obtained for the hectemoria referred to the 

 pole (VIII). We might here remark, that, in that equation, the arc D' is 

 projected upon a plane touching the sphere at the pole, into its tangent, 

 and the angle L into an equal angle. That equation is, then, the equation 

 of the projected hectemoria upon such a plane ; and we are led to the same 

 result by the equation which we have just obtained for the cone, the cone 

 having merged into the tangent-plane when the latitude of its circle of con- 



