upon the Antique Dials. 107 



but they are not in general the most simple. An exception to this happens 

 when K = 0, or the plane touches the equator, as will presently appear. 



The intersection of oc with the tangent plane, offers a result nearly the 

 same in form as that given by EULER'S formulae of transformation ; and, 

 therefore, had this transformation been the most convenient, we might 

 have adopted that formulae at once. It is wanting, however, in symmetry, 

 and is, besides, less simple than one yet to be noticed. The same objection, 

 but in a stronger degree, exists against adopting the intersection of y, with 

 the tangent plane for the origin. 



There only remains, then, the intersection of z, and we proceed to 

 give the results of taking this for the origin. The value of the addends 

 are, 



if" = a cosec A, 



Making these substitutions, and a few simple reductions, we arrive at 

 length to 



x' cos A + si sin A + a cosec A 



I (a/ sin A + z' cos A)* + y* | * 



Xcot I =r 



i 

 = cosn cos 



a? sinA + 2' cosA)* 



which is the equation of the hectemorial cone referred to its axes a^ y' z', and 

 originating at the intersection of z with the plane of projection. 



XXIII. 



To find the equation of the temporary hour-lines upon this plane, we 

 have only to put z' = 0, and efface the accents from x',y'. Thus we 

 have 



x cos A 4. a cosec. A - 1 x sin 7 cos 1 4- y sin I 



, . cot I = cos n cos . 9 (A c u\. 



! m 2 A 248 y) 



